3414
J. FERIZY, R. MANCKE, E.
~ I A E K A w A , Y. OY.4NAG1, AND
R. DICKIE
Dynamic Mechanical Properties of Cross-Linked Rubbers. I. Effects of Cross-Link Spacing in Natural Rubber‘
by John D. Ferry, Ralph G. Mancke, Etsuji Maekawa, Yasuji Oyanagi, and Ray A. Dickie Department of Chemistry, University of Wisconsin, Madison, Wisconsin (Received August 8,1964)
The complex shear compliances of five samples of natural rubber cross linked by dicumyl peroxide and three samples cross linked by sulfur, covering a broad range of average crosslink spacing, have been measured over a frequency range of 0.1 to 1000 c.p.s. and a temperature range from - 18 to 55”. The data were all reduced to 25” by shift factors calculated from an equation of the WLF type. In the transition zone of frequencies, the viscoelastic functions were closely similar ; the loss tangents of the dicumyl peroxide vulcanizates were identical within experimental error, and those of the sulfur vulcanizates were shifted somewhat to the left on the logarithmic frequency scale with increasing degree of vulcanization. In the rubbery zone, the losses persisted to much lower frequencies than could be expected from configurational rearrangements within individual strands on the basis of current molecular theories, reflecting slow relaxation mechanisms whose presence has also been deduced from other measurements. The magnitude of the loss tangent a t a given frequency in this zone (e.g., 1.6 c.P.s.) increases substantially with the average spacing between cross links in both dicumyl peroxide and sulfur vulcanizates. The possible role of trapped coupling entanglements as a source of the slow relaxation mechanisms is discussed.
Introduction WIeasurements of dynamic viscoelastic properties of cross-linked as well as of stress relaxation,5,6 have revealed slow relaxation mechanisms with relaxation times much too long to attribute to configurational rearrangements within individual network strands on the basis of current molecular theories. A satisfactory explanation of these mechanisms has not been achieved. In dynamic measurements, they are manifested by mechanical losses which persist to very low frequencies. In an earlier s t ~ d y we , ~ noted differences between natural rubber vulcanizates cross linked by sulfur and by other agents such as dicumyl peroxide or y-radiation and inferred that the chemical nature of the cross link was important in determining the low-frequency losses. However, the spacing between cross links varied somewhat in that series of vulcanizates. We have now studied two series of samples, cross linked by dicumyl peroxide and by sulfur, respectively, with different The Journal of Physical Chemistry
cross-link spacings. The average molecular weight of the network strands is found to have an important influence on the low-frequency losses.
Experimental Materials. The cross-linked rubbers were generously prepared for us in the laboratory of Dr. P. Thirion, Inst,itut Franqais du Caoutchouc, Paris ; identical preparations have been used in some of his stress relaxation studies.7r8 The dicumyl peroxide vulcanizates, ~~
~
(1) Part XLVII of a series on mechanical properties of substances of high molecular weight. (2) W. Philippoff, J . A p p l . Phys., 24, 685 (1953). (3) A. R. Payne in “Rheology of Elastomers,” P. Mason and N. Wookey, Ed., Pergamon Press Ltd., London, 1958, p. 86. (4) R. A. Stratton and J. D. Ferry, J . Phys. Chem., 67, 2781 (1963); Rev. gen. caoutchouc, 41, 635 (1964).
(5) P. Thirion and R. Chasset, Proceedings of the Fourth Rubber Technology Conference, London, 1962, p. 338. (6) A. N. Gent, J . A p p l . Polymer Sci., 6, 433 (1962). (7) P. Thirion, “Proceedings of the Fourth International Congress on Rheology,” Interscience Publishers, New York, N. Y . , in press.
3415
DYNAMIC MECHANICAL, PROPERTIES OF CROSS-LINKED RUBBERS _ _ _ _ ~ _ _
Series 98, all contained three parts of dicumyl peroxide (DCP) per 100 parts of smoked sheet rubber with a very low content of inorganic impurities. They were vulcanized for djffermt times at 135’ as shown in Tab!? I. The molecular weight between cross links, M,, w2s determined by Thirion and Chasset from swelling measurements in benzene, using a value of 0.444 for the therniodynamic parameter p ; the latter was obtained, in turn, from the swelling of samples whose M, had been calculated from tensile stresselongation measurements on two samples in a highly swollen state. The sulfur vulcanizates, Series 74, had the same composition as that numbered 74 in ref. 4: 5 parts of 2110, 1 part of phenylcyclohexyl-pphenylenediamine, 1 part of stearic acid, 3.25 parts of sulfur, and 1.25 parts of diphenylguanidine per 100 parta of smoked sheet rubber. They were vulcanized for different times at 140’ as shown in Table I, The value of J f Cis available only for sample 60 which has already been described in ref. 4. The equilibrium shear compliances, J,, were measured in the torsion pendulum as previously described4 with a maximum shea,r strain of about 0.007 and are also given in Table I. In view of the slow relaxation processes described by Thirlon,7,*these may be too low by several per cent, even though the samples were equilibrated under stress for several hours; for the samples identified as F and G, the error is probably ~~
Table I : Characteristics of Cross-Linked Rubbers
A4gent
DCP (98)
Sample codea
F G
R I J
8 (74)
Vulcanisation time, min.
20 30 40 75 I. 50 30 45 60°
iM, x IO-:
20.0 14.8 11.4 7.6 5.4
... ...
4.7
Log J e from direct shear Log J, from meaaureequilibrium extension,* menta, crn.s/dyne cm.Z/dyne
-6.36 -6.42 -6.40 -6.58 -6.65 -6.54 -6.63 -6.70
-6.29 -6.41 -6.51 -6.65 -6.75
Identification used in ref. 7 and 8. *From data of ref. 7 and 8. Sample 74 of ref. 4.
greater. An alternative calculation of J , is obtained from the stress relaxation data of Thirion,’>8taking the extrapolated ratio of F e / ( h to vanishing tensile strain as one-third of Young’s modulus E,, and using the relation J , = 3/Ee. Here P,, the equilibrium force per unit area at a given extension ratio A, is
obtained by an empirical analysis of the relaxation f u n c t i ~ nwhich ~ , ~ amounts to extrapolation to infinite time. In the DCP series, M, varies over a factor of 3.7, but the range of J , is somewhat smaller; this difference is associated with well-known deviations from the simple theory of rubber-like elasticity, as a result of which J , and M , are not directly proportional. However, J , is an indirect measure of the spacing between cross links, and in the sulfur series it is the only measure available. ~ Fitzgerald Methods. As in previous ~ t u d i e s ,the transducer was used for measurements of the storage (J’) and loss (J”) components of the dynamic shear compliance in the range from 30 to 1000 c.P.s., and the Plazek torsion pendulum in the range from 0.1 to 1.3 C.P.S. The logarithmic decrement was determined from the recorder tracings of the torsion pendulum by a method of direct matching to exponential envelopes which will be described elsewhere. The maximum temperature range was from -18 to 55’. The more highly cross-linked samples could be taken to lower temperatures without encountering crystallization; any data showing 6he symptoms of crystallization (drop in magnitude of J’ and J”, failure to superpose with reduced variables at different temperatures) were rejected. The disk-shaped samples were 1.75 cm. in diameter; the thickness was about 0.13 em. for the transducer measurements and 0.64 cm. for the torsion pendulum. Since the question had been raised in discussion with other investigators as to whether friction effects at the interface between sample and either the top or bottom plate of the torsion pendulum could influence the observed damping, a series of measurements was undertaken on several samples of the same rubber with different thicknesses. The rubber was a sulfur vulcanizate denoted as 77 in ref. 4, which had been stored at -5’ since the measurements by Stratton 3 years previously. A single disk, 0.218 cm. thick, was used, and combinations of two and three such disks layered on each other, as well as a combination of one such disk with a thinner one to give a total thickness of 0.297 cm., were also taken. A very slight tackiness provided satisfactory adherence of the disks to each other and to the upper and lower torsion pendulum plates. The disks were compressed in situ about 2.4% in every case. Measurements taken at 15, 25, and 3j0, reduced to 25’ by the method of reduced variables as described below, agreed very closely with each other, ~
(8) R. Chasset and P. Thirion, “Proceedings of the International Conference on Non-Crystalline Solids,” S o r t h Holland Publishing Co., Amsterdam. in press.
Volume 68. Number 1 I
November, 1964
J. FERRY, R. MANCKE, E. MAEKAWA, Y. OYANAGI,AND R. DICKIE
3416
as well as with the earlier measurements of Stratton.4 The loss tangents are plotted against the logarithm of reduced frequency in Fig. 1; a similar plot of the storage modulus showed equally good agreement. It may be concluded that there are no anomalies due to interfacial effects. Although the loss tangents measured in the torsion pendulum are sometimes very small, the background loss in the pendulum is believed to be far smaller; its oscillatory behavior with no sample indicates that the background is never more than 1% of the magnitude measured with the samples described here.
Results For plotting with reduced variables, the storage and loss compliances, J f and J”, were reduced in magnitude by the usual factor Tp/Topa, where p and po are the densities at the temperature of measurement T and the reference temperature of To = 298.2’K.; the application of this factor is denoted by the subscript p. For this purpose the thermal expansion coefficient of the rubber was taken as 7.1 X lo-* deg.-’. The frequency shift factor UT was calculated by the equatione log UT
=
-5.94(T - 298.2)/ (151.6
+ T - 298.2)
* Z O * O0
0.5
0
1.5
1
log w a T Figure 1. Loss tangent plotted against logarithm of reduced frequency a t 25“ for sulfur vulcanizate 77 of Stratton and Ferry4 for different sample thicknesses as follows: pip left, single disk, 0.218 cm.; pip down, two disks, total 0.297 cm.; pip right,.two disks, total 0.433 cm.; pip up, three disks, total 0.646 cm. Points include measurements a t 15, 25, and 35’. Curve is reproduced from earlier measurements by Stratton in 1961.
I
‘ 30 -
-
-
I
I
~ - -
“
150
I
(1)
In Fig. 2, J’ and J” are plotted logarithmically in this manner for two dicumyl peroxide vulcanizates with low (30) and high (150) cross-link densities. The low-frequency data from the torsion pendulum and the high-frequency data from the transducer match satisfactorily for J’, indicating that the sample coefficients based on sample geometry are correctly determined (as is not always the case4). For J”, the low-frequency losses appear higher in the transducer, but these are believed to be less reliable because the driving tube loss correction is substantial when the loss is so small, and there is no known source of a negative error in the loss measured by the torsion pendulum. Hence the J”, curves at low frequencies have been based on the torsion pendulum measurements. In comparing the two samples, that with the shorter spacing between cross links (150) has a somewhat smaller J’ at low frequencies, as would be expected, and a much smaller J”, indicating that the unexplained low-frequency losses diminish with reduction in cross-link spacing. The same conclusion is reached by comparing the sulfur vulcanizates. In Fig. 3, J’ and J” for sample 60 are reproduced from ref, 4; the data for the other two samples with larger cross-link spacings show larger values of J” at low frequencies. The Journal of Physical Chemistry
0.06
150
- 9.0 -1
I
I
0
1
I
2 log w a T
I
I
I
3
4
5
Figure 2. Storage and loss compliance reduced to 25’ and plotted logarithmically against reduced radian frequency for two samples cross linked by dicumyl peroxide, identified by vulcanization times (Table I). Pips denote different temperatures, spaced a t about 5” intervals from -6.7 (sample 30) and -17.9 (sample 150) to -5O, and then 10’ intervals from -5 to 54.5”.
The most direct measure of relative loss is the loss tangent, tan 6 = J”/J‘. This quantity is plotted logarithmically against frequency for all the dicumyl peroxide vulcanizates in Fig. 4 and the sulfur vulcanizates in Fig. 5 . At low frequencies the level of tan 6 falls markedly with increasing degree of vulcanization.
Discussion The Transition Zone. Although the primary interest of this study is the rubbery zone at low frequencies, it
DYNAMIC MECHANICAL PROPERTIES OF CROSS-LINKED RUBBERS
0'
-6.5
3417
I
1
0
I
1
I
I
4
5
3
w
-
-?.O
-7.5 3
w -8.0
-0
-1
0
I
3
2
4
5
6
109w a T
Figure 3. Storage and loss compliance reduced to 25" and plotted logarithmically against radian frequency for three samples cross linked by sulfur, identified by vulcanization times (Table I). Pips denote different temperatures, spaced a t about 5" intervals from -12.2 to 5.3" and then 10" intervals to 35".
c
-0.5
i/c E U i-
-r-1.c
-1.5
-2.0
loq waT a t
25'
Figure 4. Loss tangents of dicumyl peroxide vulcanizates plotted logarithmically against radian frequency reduced to 25'. Numbers denote vulcanization times in Table I.
may be remarked that in the transition zone where tan 6 becomes of the order of unity the position on the
logarithmic frequency scale is determined primarily by the local friction coefficient which will be more sensitive to differences in chemical constitution than to the degree of cross linking. I n agreement with this expectation, all the dicumyl peroxide vulcanizates, for which the cross-linking process is unaccompanied by side reactions influencing the composition of the polymer, converge to a single curve for tan 6 as the transition zone is entered.Q On the other hand, sulfur vulcanization is accompanied by side reactions which increase the friction coefficient a t a given temperaturelo; the
-2.0 2 100
3
w a y a t 25'
Figure 5. Loss tangents of sulfur vulcanizates plotted logarithmically against radian frequency reduced to 25". Numbers denote vulcanization times in Table I.
position of the curve on the frequency scale in the transition zone is, accordingly, shifted to the left with increasing degree of vulcanization (Fig. 5). The Rubbery Zone. At present, the complete effect of cross-link spacing on losses In the rubbery zone in Fig. 4 and 5 cannot be'adequately interpreted because there is no guiding theory. The Rouse theory,ll modified for network^'^,^^ and taking into account strand length di~tribution,~ predicts a rapid decrease in tan 6 with decreasing frequency and no curvature convex to the abscissa axis. For a partial quantitative description, the value of log tan 6 at a fixed frequency of 10 radians/sec. or about 1.6 C.P.S. is plotted in Fig. 6, both against log M , (dicumyl peroxide series only) and against log J , as an alternative measure of cross-link density (both series). It is evident that the effect of cross-link spacing in determining the magnitude of the loss is much more important than that of the chemical nature of the cross link. Possible Role of Entanglements. It is suggestive that the average spacing between coupling entanglements in the polymer before cross linking, Me', is in the range of M , covered in these vulcanizates. Only a very rough estimate of Me' can be obtained for (9) Measurements of diffusion through rubber networks (S. P. Chen, unpublished experiments) indicate that there is a small dependence of the friction coefficient on density of cross linking, insufficient to be clearly distinguished in Fig. 4. (10) H. D. Heinie, K. Schmieder, G. Schnell, and K. A. Wolf, Kuutachuk Cummi, 14, 208 (1961). (11) P. E. Rouse, Jr., J . Chem. Phys., 21, 1272 (1953). (12) M. Mooney, J . Polymer S e i . , 34, 599 (1959). (13) R. B. Blizard, J. A p p l . Phys., 22, 730 (1951).
Volume 68, Number 11
November, IS&.
3418
J. FERRY, R. RIASCKE,E. MAEKAWA, 'I-OYAKAGI, . A N D R. DICKIE
3.8
I
I
4.0
4.2
Log M, (swelling)
I
I
-6.6
-6.4
Log Je
Figure 6. Loss tangent a t log WCLT = 1, plotted logarithmically against molecular weight between cross links determined from swelling (left) and equilibrium compliance (right). Open circles, dicumyl peroxide vulcanizates ; black circles, sulfur vulcanizates. Tags denote data from ref. 4.
natural rubber, froin the pseudo-equilibrium plateau in the modulus of a linear polymer of high molecular weight14; it is 5000 to 7000. In analyzing the relation between the density of physically effective cross links and the density of cheinical cross links introduced in the vulcanization process, ;\lullinsl~has deduced that trapped entanglements with a molecular weight spacing of about 14,000 contribute to the equilibrium niodulus in natural rubber vulcanizates. The treatment tacitly identifies these entanglements with coupling loci already present before cross linking rather than the topological entanglen~elits caused by the cross linking as discussed by F10ry.l~ A similar treatment of entanglements in polybutadiene networks has been given by Kraus.17 It can hardly be expected that exact agreement between the entanglement spacings in the linear and cross-linked polymers would be obtained, even if they do represent the same loci, since both estimations are very rough; the network calculations may be subject to several corrections.18l 9 The very long relaxation times revealed by the lowfrequency losses may thus be associated with readjustnients of configurational distributions coupled through trapped entanglements and extending over relatively long distances through the network volume. When
The Journal of Physical Chemistry
Ad, > X e ' ,as is ahnost certainly the case for sample 20 of the DCP series, some strands will have two or more entanglements arid there will be a substantial nurn ber of such coupling sequences. When Mc < M e f , as in sample 150 of the DCP series, many of the strands will have no entanglements, and such sequences will be rare. Qualitatively, this picture is in accord with the dependence of losses on cross-linking spacing, but a inore precise forniulation is needed. The probable role of trapped entanglements in slow relaxation niechariisnis in rubber networks has also been deduced by Kraus20 from stress relaxation experiments and by PlazekZ1 from creep. There are, however, other possible sources of slow relaxation mechanisms, such as network strands with free ends and unattached molecules ("sol fraction"). These will be investigated subsequently. The relation of our dynamic measurements to transient measurements by Thirion on samples of identical coniposition will be discussed in more detail elsewhere.
Acknowledgment. R e are grateful to Drs. P. Thirion and R. Chasset for preparation of samples, examination of their unpublished data, and valuable discussions. This work was supported in part by the U. S.Army Research Office (Durham), in part by the Sational Science Foundation, and in part by the Research Committee of the Graduate School of the University of Wisconsin. R e are indebted to Miss Nonona Rossol and Miss Marilyn Etzelmueller for help with calculations. (14) H. Markovita, T. G Fox, and J. D. Ferry, J . Phys. Chem., 66, 1567 (1962).
(15) L. AMullins.J . A p p l . Polymer Sci., 2, 1 (1959). (16) P. J. Flory, "Principles of Polymer Chemistry," Cornel1 University Press. Ithaca. N. Y . , 1953, pp. 461-464. (17) G. Kraus, J . A p p l . Polymer Sci., 7, 1267 (1963). (18) J. d. Duiser and A . J. Staverman, "Proceedings of the International Conference on Xon-Crystalline Solids," North Holland Publishing Co.. Amsterdam, in press.
(19) W. Prins, ibid.. in press. (20) G. Kraus and G. A. Mocaygemba. J . Polymer Sci., A2, 277 (1964). (21) D. J. Plazek, private communication.
