RAYA. DICKIEAND JOHND. FERRY
2594
Dynamic Mechanical Properties of Cross-Linked Rubbers. 111. Dicumyl Peroxide Vulcanizates of Natural Rubber1
by Ray A. Dickie and John D. Ferry Department
of
Chemistry, Unive~sityof Wisconsin, Madison, Wisconsin (Received March 7, 1966)
Viscoelastic properties of ten samples of natural rubber, prepared from stocks of four different initial molecular weights and cross-linked by dicumyl peroxide to different extents, have been studied. The complex shear compliances were measured over a frequency range from 0.2 to 2 cps at temperatures from -25 to 50" by a torsion pendulum. On three of the samples, measurements were extended by the Fitzgerald transducer from 45 to 600 cps. Creep measurements were made over time periods up to 1 day from -15 to 35" on all ten samples as well as five additional natural rubber vulcanizates and one lightly cross linked polybutadiene for which dynamic data have been previously reported; the creep data were converted to the corresponding dynamic viscoelastic functions a t very low frequencies by conventional approximation methods. All data were reduced to 25" by shift factors calculated from a previously adopted equation of the WLF form. Equilibrium compliances were calculated from the creep measurements by a modification of the empirical equation of Thirion and Chasset for stress relaxation. The equilibrium compliance was inversely proportional to the network strand density (uncorrected for free ends) as obtained by Thirion and Chasset from swelling measurements. For the samples with least cross linking, an inflection in the storage compliance and maxima in the loss Compliance and loss tangent a t very low frequencies revealed the presence of a secondary loss mechanism as previously found in lightly cross-linked polybutadiene. The contribution of this mechanism to the equilibrium compliance was estimated by integrating over the retardation spectrum. The magnitude of the low-frequency loss mechanism, as measured by either the value of the loss tangent a t the frequency where it. is most prominent or the value of the retardation spectrum at the point on the time scale where it is most prominent, is closely correlated wit,h the equilibrium compliance, in a monotonic function in which the separate effects of free ends and chemical cross-link density cannot be distinguished.
Introduction I n earlier papers, we described dynamic mechanical measurements on natural rubber2 and p~lybutadiene~ cross-linked by dicumyl peroxide and by sulfur. Anomalous losses were observed a t low frequencies, especially in networks with low degrees of cross linking. I n the present study, a more extended series of natural rubber vulcanizates is treated; both the initial molecular weight and the degree of cross linking by dicumyl peroxide have been varied, in an attempt to distinguish their respective effects on the low-frequency losses. The dynamic mechanical measurements have been The Journal of Physical Chemdstry
supplemented by creep measurements to provide the retardation spectrum over an extended range of time scale in the region of the secondary loss mechanism.
Experimental Section Materials. The cross-linked rubbers were generously prepared for us in the laboratory of Dr. P. Thirion, (1) Part L of a series on mechanical properties of substances of high molecular weight. (2) J. D. Ferry, R. G. Mancke, E. Maekawa, Y. Gyanagi, and R. A. Dickie, J . Phys. Chem., 68, 3414 (1964). (3) E. Maekawa, R. G. Mancke, and J. D. Ferry, ibid., 69, 2811 (1965).
DYNAMIC MECHANICAL PROPERTIES OF CROSS-LINKED RUBBERS
Institut Franpais du Caoutchouc, Paris. Portions of a smoked sheet natural rubber specially selected for high purity were masticated for different times with 3.5 parts of dicumyl peroxide/100 parts of rubber, and subsequently divided into subportions which were vulcanized at 135" for different times. A control sample, masticated under identical conditions without peroxide, was used in each case for determination of the initial molecular weight before vulcanization. The latter value was calculated from the intrinsic viscosity [ q ] measured in cyclohexane a t 25" by the equation of Altgelt and Schulz4
M
=
200[~11.4
(1)
The average spacing between cross links, M,, was determined in Dr. Thirion's laboratory from swelling measurements in benzene, as described earlier.2 From this, the physical network strand density was calculated as v = p / M , , where p is the density. The values of M , and v contain no correction for free ends of the form 1 - constant/M, since v is to be interpreted as the effective strand density uncorrected for any network defects. The densities were determined pycnometrically; they ranged from 0.911 for the least to 0.915 for the most highly cross-linked sample. The characteristic parameters for the vulcanizates are given in Table I, together with data for some dicumyl peroxide vulcanizates previously studied2 which will be included in the subsequent analysis. Table I : Characteristics of Cross-Linked Rubbers
.w
Sample code
10-8 before vuloanization
Vulcanization time, min
Ai-20 Ai-40 Az-40 Az-75 A2-150 B1-20 BI-40 Bz-40 Bz-75 Bz-150
550 550 770 770 770 260 260 230 230 230
F G H
20 40 40 75 150 20 40 40 75 150 20 30 40
I J
150
75
2595
ponents of the dynamic shear compliance in the range from 45 to 600 cps for three samples. The Plazek torsion pendulums was used in the range from 0.2 to 2 cps for all ten samples. The maximum temperature range was from -25 to 50". The disk-shaped samples were 1.75 cm in diameter; the thickness was about 0.19 cm for the transducer measurements and 0.64 cm for the torsion pendulum. In the torsion pendulum measurements, the logarithmic decrement was determined from the recorder tracings of the angular displacement by direct matching to exponential envelopes. The recorder speed was adjusted (with the help of a speed reduction adapter) so that the decaying oscillations neither fell to noise level in less than 4 or 5 cm nor remained larger than onehalf the initial value after 12 cm. A chart paper' was used on which the trace appears as a translucent line. The recording was projected through an automatic focusing photographic enlarge? on a large sheet of accurate graph paper, on which had been drawn seven pairs of standard exponential curves corresponding to y = *A,e-ksZ with A , about 10 cm and values of k, ranging from 0.0181 to 0.200 cm-l. By continuous variation of the degree of enlargement, the experimental curve could be precisely located between the envelopes of one (or another) of the pairs of standard curves. At this position, the distances between several maxima and minima were measured on the graph paper scale and averaged ( = P ) ; also, the average distance between time-interval blips marked by the recorder ( = tP) was determined. Then the logarithmic decrement A and angular frequency of oscillation (J are giveng by A = k.P
10-aMO
104~
19.4 12.2 11.2 7.2 4.9 31.8 15.7 16.9 10.3 6.0 20.0 14.8 11.4 7.6 5.4
0.47 0.75 0.81 1.27 1.86 0.29 0.58 0.54 0.88 1.52 0.46 0.62 0.80 1.20
1.69
Methods. The Fitzgerald transducer5 was used for measurements of the storage ( J ' ) and loss (J") com-
w =
27rtp/P
It is unnecessary to know the magnification factor explicitly. This method was found to be a t least a,s precise as and far more rapid than the previous procedure from plotting coordinates of maxima and minima on a logarithmic scale. The torsion apparatus was also used for measurements of creep. The arrangement described earlier1° ~
~
(4) K. Altgelt and G. V. Schulz, Makromol. Chem., 36, 209 (1960). (5) E. R. Fitzgerald, Phys. Rev., 108, 690 (1957). (6) D. J. Plazek, M. N. Vrancken, and J. W. Berge, Trans. SOC. Rheol., 2, 39 (1958).
(7) Sanborn Permaper No. 615-181. (8) Simmon-Omega Automega Model D-3 with a 135-mm f 3.5 lens, capacity 4 X 5 in. (9) R. A. Dickie, Ph.D. Thesis, University of Wisconsin, 1965. (10) K. Ninomiya, J. R. Richards, and J. D. Ferry, J. Phys. Chem., 67, 327 (1963).
Volume 70,Number 8 August 1966
RAYA. DICKIE AND JOHND. FERRY
2596
for measuring small angular displacements was modified by using a pair of silicon solar cells centered on the edges of the light beam; their outputs were opposed to give a null galvanometer reading corresponding to the location of the beam axis.ll This greatly improved the precision and reproducibility. The creep measurements were extended to a maximum of about 2 days; the maximum shear strain was about 0.002. Measurements were made at temperatures from - 15 to 35”.
-6.2
Results Both creep and dynamic data were reduced to a reference temperature of 25’ in the usual manner, taking the thermal expansion coefficient of the rubber as 7.1 X lo-* deg-l and the shift factor CZTfrom the equation log UT = -5.94(T - 298.2)/(151.6
+ T - 298.2) (3)
as in earlier studies on natural rubber.2*12 Creep and Equilibrium Compliance. Reduced creep curves are shown in Figure 1 for the two samples of Table I with the least cross linking and also for the least cross-linked polybutadiene vulcanixate (Code 734) for which dynamic measurements have been previously r e p ~ r t e d . ~Here the creep compliance J ( t ) is plotted logarithmically against the reduced time UT. Similar curves for the other samples are given in the thesisg on which this paper is based; with increasing degree of cross linking, the magnitude of the retarded compliance diminished, the slope being very small for the most highly cross-linked samples. I n view of the improved precision of the apparatus, five dicumyl peroxide vulcanixates of natural rubber which had been the subject of an earlier paper2 were also subjected to creep measurements, primarily to determine their equilibrium compliances with greater accuracy. The equilibrium compliance obviously cannot be estimated by inspection of the curves in Figure 1, and, even for the most highly cross-linked samples, equilibrium was not quite attained within the experimental time scale. Recourse was therefore taken to extrapolation by an empirical relation analogous to that used by Thirion and C h a ~ s e t ~ ~for . l * stress relaxation, which can be put in the form
G(t) = Ge[1
+ (t/tm>-ml
(4) where G(t) is the relaxation shear modulus a t time t, G, is the equilibrium modulus, and t, and m are empirical constants. This expression has been found to fit slow relaxation data on a variety of cross-linked l 4 From the phenomenological rubbers very theory of viscoelasticity,1Kfor a time dependence as The Journal of Physical Chemiatry
-I
0
I
I
I
I
2
3
4
5
Ioq t/a,
Figure 1. Creep compliance for the two natural rubber vulcanizates with least cross linking, identified by values of Y and the polybutadiene code 734 (PB), all reduced to 25”. Temperature key (approximate): pip up at short times, -15”; successive 45” rotations clockwise denote 0, 5, 10, 15, 20, 25, 30, 35”. Exact temperatures are given in ref 9.
slight as shown in Figure 1 , J ( t ) is within 1% simply the reciprocal of G(t). Hence, we may write
J(t)
=
Je/[1
+ (t/tm)-ml
(5)
which can be rearranged to give log [ J e / J ( t )- 11 = -m log t
+ m log 1,
(6)
By trial and error, a value of J , can be chosen which produces a linear plot of the left side of eq 6 against log t. The parameters m and t, are also determined, and these are listed together with log J e in Table I1 for the samples of Table I as well as the dicumyl peroxide vulcanixates of ref 2. For the latter, log J , had been previously derived from the stress relaxation measurements of Chasset and Thirion;14 these values are also given in Table I1 together with values obtained recently by Plazek16 from a quite different type of analysis based on creep measurements in his very precise magnetic-torque instrument. l7 All three results are in fairly good agreement. If the equilibrium compliance reflected the same physical network strand density as the swelling, subject (11) We are indebted to Dr. D. J. Plazek for this suggestion. (12) R. A. Stratton and J. D. Ferry, J . Phys. Chem., 67, 2781 (1963). (13) P. Thirion and R. Chasset, Rev. Gen. Caoutchouc, 41, 271 (1964). (14) R. Chasset and P. Thirion, “Proceedings of the International Conference on Non-Crystalline Solids,” North-Holland Publishing Co., Amsterdam, 1965, p 345. (15) J. D.Ferry “Viscoelastic Properties of Polymers,” John Wiley and Sons, Inc., New York, N. Y.,1961. p 71, eq 41. (16) D.J. Plazek, J. Polymer Sci., in press. (17) D.J. Plazek, Trans. SOC.Rheol., 7 , 61 (1963).
DYNAMIC MECHANICAL PROPERTIES OF CROSS-LINKED RUBBERS
2597
Table 11: Equilibiium Compliances and Parameters Describing Creep (All Reduced to 25") -Log J e (in cm*/dyne,
Sample code
-Log J e -Log J e (creep's)
creep)
Ai-20 Ai-40 An-40 A2-75 Az-150 B1-20 BI-40 Br40 Bz-75 Bz-150
6.19 6.46 6.47 6.62 6.80 5.98 6.37 6.41 6.52 6.77 6.28 6.35 6.42 6.57 6.68
F G H I J
6.23 6.36 6.46 6.61 6.72
(stress relaxation)
6.29 6.41 6.51 6.65 6.75
m
m log tm
tm
x x x x
0.073 0.184 0.054 0.104
1.6 1.7 2.0 2.4
0.102 0.149 0.066 0.185 0.382 0.0925 0.115 0.160
5.8 x 5.0 x 4.2 x 4.0 x 3.6 x 1.6 x 1.4 x 4.0 x
10-8 10-10
10-lo 10-l 10-11 10-7 10-4 10-4 10-4
-0.132 -0.510 -0.526 -1.00 -0.024 -0.641 -0.682 -1.184 -1.314 -0.259 -0.443 -0.544 -0.977' - 1.266"
-../L
' From analysis of stress relaxation: private communication from Dr. P. Thirion.
\
-40
'\
x
-
-4.2.
-4.4.
..-
109
- 6.8
-6.6
- 6.4 109
-6.2
-6.0
Je
Figure 2. Effective physical network strand density plotted logarithmically against equilibrium compliance a t 25" : pip up, initial molecular weight 2.3 X 106; right, 2.6 X 106; down, 5.5 X 106; left, 7.7 X lo6; open circles, series FGHIJ, from present creep measurements; black circles, from stress relaxation by Thirion and Chasset; solid line, slope of 1; dashed line from eq 7.
-
to the same front factorls and other correction^,^^ it should be given by
Je = l/vRT (7) In fact, a logarithmic plot of Y against Je in Figure 2 shows these quantities to be directly proportional, but J, is smaller than predicted in eq 7 by a factor of
Je
Figure 3. Parameter m log tm plotted against log J,. Key same as in Figure 2.
0.71. A similar discrepancy has been found for polybutadiene and in other studies of natural rubber.20p21 It is of interest to examine the empirical parameters m and 1,. Their individual dependences on v or J, are somewhat erratic, because there is some latitude in the selection of either m or tm singly; this latitude largely disappears in the quantity m log t,, which is (18) A. V. Tobolsky, D. W. Carlson, and N. Indictor, J . Polymer Sci., 54, 175 (1961).
(19) J. D.Ferry, "Proceedings of the International Conference on Non-Crystalline Solids," North-Holland Publishing Co., Amsterdam, 1965,p 333. (20) L. E. Nielsen, J . Appl. Polymer Sci., 8 , 511 (1964). (21) P. Thirion and R. Chasset, unpublished work.
Volume 70, Number 8 August 1966
RAYA. DICKIEAND JOHN D. FERRY
2598
-801
1
-4
-3
-2
-I
I
I
0
I
1
2
3
4
I
,
5
6
l o q wai
Figure 4. Storage compliance, reduced to 2 5 O , plotted logarithmically against frequency for four natural rubber vulcanizates identified by values of Y and polybutadiene code 734 (PB). Dashed curves calculated from creep data of Figure 1. Temperature key (approximate): pip up a t high frequencies, -25"; successive 45" rotations clockwise denote successive 5" temperature increments.
____________----------........................
P
Y
0-
-0
loq w a T
Figure 5. Loss compliance, reduced and plotted
&s
in Figure 4, with same key.
plotted against log J, in Figure 3. Most of the points fall on a monotonically increasing curve which, except for the most lightly cross-linked samples (highest J,), haa a slope close to 2.2. The physical significance of tmm is the relative unrelaxed stress, in excess of the equilibrium value, in a stress relaxation experiment at an elapsed time of 1 sec. Dynamic Measurements. I n Figures 4 and 5, the storage and loss compliances are plotted reduced to 25" for the three lightly cross-linked vulcanizates whose creep curves are given in Figure 2, together with those for the two most densely cross-linked samples The Journal of Physical Chemistry
(A2-150 and BT-150). For the first three, curves have been extended to very low frequencies by calculating the dynamic properties from the creep with conventional approximation formulas.22 The storage compliance clearly shows a secondary inflection in the neighborhood of zero on t,he logarithmic frequency scale in addition to the primary transition zone in the neighborhood of 6; the same behavior appears in the loss compliance, which exhibits a secondary loss maximum. In the highly cross-linked samples, the (22) Reference 15, Chapter 4.
DYNAMIC MECHANICAL PROPERTIES OF CROSS-LINKED RUBBERS
secondary mechanism does not appear, although the magnitude of J“ a t low frequencies is still greater than would be predicted for relaxation processes within single network strands.lg Data for J’ and J” at intermediate degrees of cross linking are presented el~ewhere.~ The loss tangent, tan 6 = J ” / J ’ , is plotted logarithmically against frequency in Figure 6 for all ten natural rubber vulcanizates. The curves all converge in the transition zone as previously noted for a less complete series.2 The magnitude of the low-frequency loss decreases with increasing Y, the effective physical strand density, and with one exception the relation is monotonic; the effect of initial molecular weight, on this basis, cannot be clearly distinguished.
2599
-
-2.01.52
..I
1.86
-4
Discussion
-2
1
I
I
I
0
2
4
6
lo9 w a T
Retardation Spectrum. From the data of Figures 1, 4, and 5, the retardation spectra L were calculated by conventional approximation formulaszzand are plotted in Figures 7 and 8 for the four natural rubber and the polybutadiene vulcanizates, respectively. The agreement between the separate calculations from J’ and J” is in most cases excellent. Similar calculations for the vulcanizates of intermediate strand density, together with numerical values, are recorded el~ewhere.~(This calculation is an intermediate step in estimating J’, J ” , and tan 6 at very low frequencies from creep data.) The maximum in L usually associated with a network s t r ~ c t u r e ’appears ~ * ~ ~ at about - 5 on the logarithmic time scale. In addition, a secondary maximum appears several decades to the right for the samples with the least cross linking, corresponding to the lowfrequency loss mechanism. Curiously, its location on the time scale appears to be independent of the degree of cross linking. Contributions of Transition-Zone and Low-Frequencg Mechanisms to the Equilibrium Compliance. In Figure 4, the two inflections in J’ for the samples with least cross linking are not sufficiently separated to identify a plateau corresponding to the compliance associated with the transition zone alone. However, an approximate separation can be achieved by integrating under the maxima in Figure 7 in accordance with the relation
Figure 6. Loss tangent for natural rubber vulcanizates, plotted logarithmically against frequency reduced to 25’. Key to initial molecular weights: dots, 2.3 X lo6; dot-dash, 2.6 X los; dash, 5.5 X lo6; solid curve, 7.7 X lo6. -7
7 -9
I
I
-. 4
-6
-2
0
2
4
lo9 T
Figure 7. Retardation spectra of four natural rubber vulcanizates from data of Figures 1, 4, and 5. Circles with top black, from J ’ ; bottom black, from J ” ; crossed, from creep.
to the longest time t l to which the creep measurements had extended, to give J
--m
Beyond t l , it was necessary to estimate the contribution to Je from the empirical eq 5 , from which it can be showng that the contribution between r = tl and T = m l S
J, =
J
--m
Ld In r
+ J,
(8)
in which J , is negligible. For this purpose, L was plotted with a linear scale and could be arbitrarily
Jez = J e { l
-I
[1
+
m (2tl/tm)-m12 1
+ (2tl/tm)-m (10)
Volume 70, Number 8 August 1966
RAYA. DICKIEAND JOHN D. FERRY
2600
-7.5-
?
0 I
-6
I
I
-4
-2
I
109
IO
2
0
4
-8.0-
F
T
-
0 W
Figure 8. Retardation spectrum of polybutdiene vulcanizates from data of Figure 1 and ref 3. Key same as in Figure 7.
t
rJ -I
-8.5-
0 0-
nary transition zone is J e are summarized in Table 111.
Je1
- J,z. The results
~~
Table HI : Contributions to J , from Transition-Zone
I
and Low-Frequency Mechanisms
-6.8
-Sample Bi-20 1 0 4 ~
1 0 7 ~ ~ ~ 1 0 7 ~ ~ ~ Low-frequency mechanism x 107 Transition zone X lo7 107~.
0.29 4.8 2.9 7.7 2.7 10.4
-6.4
I
-6.2
I
-6.0
109 Je
codAd0
Figure 9. Magnitude of retardation spectrum at log -0.5 (open circles) and loss tangent at log w = 1.4 (black circles), plotted logarithmically against J,. Key to initial molecular weight same as in Figure 2.
0.47 2.2 1.9 4.1 2.4
6.5
It is evident that considerably more than half of the total equilibrium compliance is associated with the lowfrequency mechanism. This reflects structural features which support stress on a time scale of the order of 1 sec but relax at much longer times. If these are entanglements, they must be entanglements whose effects vanish at equilibrium. It is striking that the effects extend to such long times and also that the retardation spectrum is so broad. Correlation of Low-Frequency Losses with Equilibrium Compliance. The best measure of the magnitude of the low-frequency mechanism would probably be Jel Je2, but, since this is unavailable for most of the samples, we have adopted two arbitrary indexes: the magnitudes of L and tan 6 near the maxima which appear in these quantities for the most lightly cross-linked samples. These are plotted logarithmically against J, in Figure 9 and with suitable choice of scales are seen to be essentially equivalent. As might have been inferred from Figure 3-since m log t, is also a measure of slow relaxation processes-and Figure 6, there is a monotonic dependence on J , (and hence, from Figure
+
The Journal of P h y e d Chemistry
-6.6
T
=
2, on v) in which differences in initial molecular weight are not distinguishable. It has previously been c ~ n c l u d e dthat ~ ~ ~the lowfrequency losses increase with decreasing initial molecular weight (increasing proportion of loose-end strands) and decreasing density of chemical cross links. However, J e also depends in the same manner on both of these variables, so their effect cannot readily be separated. To do so would require knowledge of the chemical cross-link density. Although this quantity has been derived from v previously on the basis of certain assumptions about entanglements, 3, l9 reexamination of these assumptions has thrown doubt on their validity.16u21 A further analysis of the data of this and preceding papers in the light of this reexamination will be presented in a subsequent communication.
Acknowledgment. This work was supported in part by the U. S. Army Research Office (Durham), in part by the National Science Foundation, and in part by the Research Committee of the Graduate School of the University of Wisconsin. We are grateful to Drs. P. Thirion and R. Chasset for furnishing the samples and for many valuable comments in the course of this work. We are also indebted to Misses Monona ROSSOI, Marilyn Etzelmueller, and Janet Gomoll for help with calculations.
