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Constant stress elongation of soft polymers: Time and temperature studies
C O N S T A N T S T R E S S E L O N G A T I O N OF S O F T P O L Y M E R S : TIME AND TEMPERATURE STUDIES 1 C. A. Dahlquist and M. R. Hatfield Central Research Department, 2 Minnesota lYIining and Manufacturing Company, St. Paul, Minnesota Received January 28, 1952
INTRODUCTION The effect of time and temperature on the viscoelastic behavior of rubbery materials is of great practical significance and has long been the subject of extensive investigation and theoretical interpretation. It has been shown that time and temperature have an equivalent effect on the elastic modulus of linear rubbery polymers (2). A quantitative relationship between time and temperature can be derived from the apparent activation energy for viscous flow (6) or for elastic deformation (3). The temperature dependence of the modulus can then be reduced to an equivalent time dependence, and data covering a wide range of time and temperature can be represented by a single "master curve" of reduced variables. The constant-stress method (5) used in this laboratory for characterization of soft polymers is well-suited for study of the time and temperature dependence of viscoelastic properties. The data are similar in many respects to the stress relaxation data of Andrews et al. (2,3,4). It was of interest to investigate the treatment of creep data by reduced variables and to compare the two methods cited above. It was also of interest to extend the treatment to polymers other than polyisobutylene, preferably to one which can be subjected to high elongation without undergoing crystallization. THEORY The modulus of elasticity, as used in this paper, is defined as the stress per unit elongation. A discussion of the applicability of such a property to constant-stress measurements on soft polymers has been given in an earlier paper (5). Reduced modulus is calculated by multiplying the experimental value by the factor 298/T, where T is the absolute temperature. This factor reduces all modulus contributions associated with entropy change to the Presented at the Annual Meeting of the Society of Rheology, Chicago, Illinois, October 24-27, 1951. 2Contribution No. 39. 253
~54
C. A. DAItLQUIST AND M. R. ItATFIELD
same reference temperature, 298°K., as predicted by the kinetic theory of rubberlike elasticity. If all retardation processes contributing to the total creep have the same temperature dependence, they c a n be characterized by a single activation energy. This activation energy can be calculated by use of the Arrhenius equation in which the reciprocal of the reaction rate at temperature T is replaced by the time, t, required to attain a given modulus. Thus t = A l e EIRT
El-1
where A t is a constant, E the activation energy, and R the gas constant. A plot of the natural logarithm of time versus the reciprocal temperature yields a straight line of slope E / R . The correction factor, K, which reduces the time corresponding to a given modulus at one temperature to that corresponding to the same modulus a t another temperature, is given by Eq. [-11 written for two temperatures K -
to t
_
e E/RT°
e EIRT"
~2~
A master curve of reduced modulus versus reduced time can be obtained by use of Eq. [21. (Andrews, Hofman-Bang, and Tobolsky in Ref. (3) give a more detailed discussion of the theory presented here.) Creep curves may be shifted along the log time axis to form a single master curve which, if the activation energy is constant, will coincide with that described above. The temperature dependence Of the time correction factor, as calculated from the actual shift, is given by the Arrhenius equation K = Ae -zIRT. [31 It should be noted that Eq. [-31, which gives the temperature dependence of K, becomes identical with Eq. E2~ if A is evaluated for the reference temperature at which K is equal to unity. A treatment based on steady-flow viscosity allows a similar reduction of time to arrive at a master curve of reduced modulus and reduced time. Recently, Ferry (6) derived the relationships which apply to dynamic properties, and the treatment for creep measurements is quite similar to that presented by Ferry. The polymeric substance is assumed to behave like a group of Voigt units in which each elasticity mechanism is represented by a modulus G~ and each flow process by riGi. Under the restriction of constant stress, the elongation at time t is given by the well-known relationship ~ = SE~(IlG~)
(1 - e-ricO,
E4~
TIME AND
TEMPERATURE
STUDIES
~
w h e r e S is t h e a p p l i e d stress. I f m o d u l u s is defined as stress p e r u n i t s t r a i n , E q . [-4] b e c o m e s 1/G, = E~(1/ai)
(1
-
e-u'i).
[5]
T h e t e m p e r a t u r e d e p e n d e n c e of all r e t a r d a t i o n m e c h a n i s m s is a s s u m e d t o b e g i v e n b y a single f a c t o r A f d e p e n d e n t o n l y u p o n t h e p o l y m e r m a t e r i a l a n d t e m p e r a t u r e . T h e m o d u l u s G~ of r u b b e r l i k e s p r i n g s is p r o p o r t i o n a l to t h e a b s o l u t e t e m p e r a t u r e so t h a t G~ = ( T / T o ) G . ,
where the subscript zero designates a reference temperature. F5] at temperature T is then
1/G,
= E~(To/T)
(1/Gi0) (1 - e-'~T~O.
[6] Equation [7]
I t follows t h a t Gt = ( T / T o ) G o u a r .
[8]
F e r r y (6) h a s s h o w n t h a t t h e s t e a d y - f l o w v i s c o s i t y n a t t e m p e r a t u r e T is given by = A T pT~?o poTo'
[9]
w h e r e p is t h e d e n s i t y . M a r v i n (8) h a s c a l c u l a t e d A T v a l u e s for p o l y i s o b u t y l e n e f r o m p r e c i s e v i s c o s i t y (7) a n d l i n e a r e x p a n s i o n (11) d a t a . T h e s e d a t a were u s e d in t h i s paper. EXPERIMENTAL The constant-stress apparatus was recently described by Dahlquist, Hendricks, and Taylor (5). A strip of polymer cut from a thin sheet or film is loaded with a weight which sinks into a liquid as the film stretches. By suitable design of the weight, which incidentally, takes the form of a hyperboloid, the load can be reduced in proportion to the change in cross-sectional area of the stretching film. The same technique has been used by Andrade (1) to study viscous flow in metallic wires. The apparatus has recently been modified to the extent that the thread release has been replaced by a magnetic release. The technique of supporting the tension weight by a thread and subsequently transferring the load to the test strip by burning the thread proved to be rather cumbersome, and a release mechanism was developed wherein the load is initially supported by an electromagnet. A soft iron rod, 1 ~ in. long X ~ in. diameter, inserted in the tab attached to the lower end of the vertically mounted test strip serves as an armature for the magnet, and supports the hyperbolic weight which is suspended by a fine wire. The load is transferred to the test strip when the armature is released by breaking the field circuit of the magnet. The tab attached to the upper end of the test strip is held in a movable clamp to permit taking up of slack in the specimen after it has been positioned in the tester. Temperature control was accomplished by enclosing the test specimen in a chamber, 5 X 4 X 14 in., with an opening at the top leading into a Dry Ice container. Air was forced through the Dry Ice and down into the test chamber, an inverted funnel just below the cold air inlet serving as a baffle to break up the air stream and distribute it uni-
256
c. A. DAHLQUIST AND M. R. HATFIELD
formly throughout the chamber. The temperature was adjusted by regulating the volume of incoming air. Temperatures above 25°C. were obtained by replacing the Dry Ice container with a preheating unit. The test strips were cut from sheets approximately 0.020 in. thick cast h'om solution on amalgamated tin panels. Polyisobutylene was dissolved in n-heptane without preliminary treatment, while GI~-S rubber received a brief cold milling. The drying schedule was as follows: (a) 24 hr. slow drying under cover, (b) 24 hr. uncovered, and (c) 16 hr. vacuum drying at 50°C. Further details of the film-casting technique are described in the earlier publication (5). The stress conditions were controlled by regulating the initial cross-sectional area of the test specimen. Had the same stress been used in all of the experiments, the deformations would have been very small at the lowest temperatures, or excessively large, leading to rupture of the specimens, at the highest temperatures. To avoid this an attempt was made to control the stress to give approximately the same 10-rain. elongation at all temperatures, ttowever, it was difficult to prepare specimens of sufficiently small cross section to fulfill the stress requirements for the -20, -35, and -50°C. tests. Since polyisobutylene tends to crystallize when extended beyond 150%, it s deformations were held below 100%. The deformation of GR-S, which does not crystallize, was permitted to exceed 200% in only one instance. At the higher temperatures, polyisobutylene exhibited a small degree of permanent set. This was subtracted from the total deformation to give the purely elastic deformation, and the modulus values were based upon the elastic component. GR-S showed no perma.hent set in these experiments. The modulus values were calculated according to the conventional "stress over strain" formula: stress × initial length G= deformation The polyisobutylene (Vistanex) was furnished by R. S. Marvin of the National Bureau of Standards.It had been carefully selected as a reference standard for comparative dynamic testing by a number of laboratories and has been described very completely by Marvin (8). It has a viscosity average molecular weight of 1.35 X 10~, a density3 of 0.913 g. cc. at 25°C., and a coefficient of linear expansion of 0.58 X 10-3/°C. GR-S X-274 was taken from a bale of commercial gum rubber. It was characterized as follows: intrinsic viscosity in benzene, 2.41; viscosity average molecular weight, 306,000 (9); Mooney viscosity, 66; gel content, zero. The measured density was 0.935. To determine the change in density with temperature the linear coefficient of expansion was taken as 1.85 X 10-4/°C., the value reported by Vieweg and Schneider (10) for Buna S-I. t~ESULTS AND DISCUSSION A b r i d g e d e x p e r i m e n t a l d a t a are p r e s e n t e d i n T a b l e s I a n d I I . Fig. i a n d 5, w h i c h c o n t a i n all of t h e d a t a , show plots of " r e d u c e d m o d u l u s " as a f u n c t i o n of t h e l o g a r i t h m of t i m e a t t h e v a r i o u s t e m p e r a t u r e s considered. T h e " r e d u c e d m o d u l u s " was d e t e r m i n e d b y m u l t i p l y i n g t h e e x p e r i m e n t a l m o d u l u s b y t h e f a c t o r 2 9 8 / T , where T is t h e a b s o l u t e t e m p e r a t u r e . If all r e t a r d a t i o n processes h a v e t h e s a m e temperat~ure dep e n d e n c e , a m a s t e r c u r v e of r e d u c e d m o d u l u s vs. t i m e m a y b e o b t a i n e d 3 Private communication from Dr. Marvin.
TIME AND TEMPERATURE STUDIES
257
b y m e r e l y s h i f t i n g t h e c r e e p c u r v e s a l o n g t h e l o g t i m e axis. T h i s i m p l i e s t h a t all t i m e v a l u e s a t a g i v e n t e m p e r a t u r e are multiplied by a single c o r r e c t i o n f a c t o r . A n d r e w s et al. (3) h a v e s h o w n t h a t t h i s c o r r e c t i o n f a c t o r is r e l a t e d t o a n a p p a r e n t a c t i v a t i o n e n e r g y . A c t i v a t i o n e n e r g y v a l u e s w e r e c a l c u l a t e d f r o m a p l o t of l o g t i m e a t constant reduced modulus vs. reciprocal temperature. Figs. 2 and 6 TABLE I Polyisobutylene Creep Data ~
Total Viscous Log thne elongation flow min. % % 1. Temp = 40°C. Stress = 2200 g./sq, cm. - 1.778 31.5 --0.699 38.6 -0.301 43.6 -0.000 49.4 0.3 0.699 70.2 2.7 1.000 86.5 7.3 - -
Elas¢.ic elonga%ion %
Log (To/T)G (G in megadynes/sq, cm.)
31.5 38.6 43.6 49.1 67.5 79.2
' 0.824 0.726 0.673 0.621 0.483 0.413
2. Temp = 10°C. -- 1.778 0.699 0.301 0.000 0.699 1.000
Stress = 3710 g./sq, cm. 43.6 -51.7 -56.0 -60.5 -75.4 0.3 85.3 0.7
43.9 52.0 56.4 60.8 75.1 84.6
0.943 0.869 0.835 0.802 0.708 0.657
3. Temp = - 2 0 ° C . -- 1.778 --1.177 --0.699 0.301 0.000 0.699 1.000
Stress = 4240 g./sq, cm. 19.2 -34.1 -41.2 -45.9 -49.0 -58.4 -61.5 --
19.2 34.1 41.2 45.9 49.0 58.4 61.5
1.408 1.158 1.075 1.029 0.999 0.924 0.900
4. Temp = - 5 0 ° C . 0.301 0:699 1.000 a Abridged table.
Stress = 7190 g./sq, cm. 24.5 -44.7 -60.3 --
24.5 44.7 60.3
1.581 1.324 1.193
- -
- -
- -
represent these plots for the two polymers under investigation. The activ a t i o n e n e r g y d e r i v e d f r o m t h e p o l y i s o b u t y l e n e d a t a (see T a b l e I V ) a p pears to be reasonably constant over the entire range, and the average v a l u e of 20.1 k c a l . is i n s u b s t a n t i a l a g r e e m e n t w i t h t h e v a l u e 19.5 o b tained by Brown and Tobolsky from stress relaxation measurements in the temperature range 0 to -45°C. (4).
258
C.A.
DAHLQUIST AND M. R. ttATFIELD
A master curve for polyisobutylene based on a constant activation e n e r g y of 20.1 k c a l . is s h o w n i n F i g . 3. T h e t r a n s l a t e d d a t a s u p e r p o s e nicely except for some scattering at the very low temperatures (short reduced time). A s e c o n d m a s t e r c u r v e b a s e d o n s t e a d y - f l o w v i s c o s i t i e s is p r e s e n t e d i n F i g . 4. T h e d a t a s h o w m o r e s c a t t e r t h a n i n F i g : 3. T h e t w o c u r v e s a r e i n TABLE II GR-S X274 Creep Data a Log time min.
Total elongationb
%
Log (To/T)G (G in megadynes/sq, cra.)
1. Temp = 40°C. --2.000 - 1:778 1.000 0.699 --0.30l 0.00O 0.699 1.000
Stress = 1306 g./sq, c a . 29.6 0.615 34.1 0.554 52.5 0.365 63.5 0.283 81.8 0.176 100.7 0.083 166.0 -0.136 208.5 --0.233
2. Temp = 10°C. 2.000 - 1.000 - 0.699 -0.301 0.000 0.699 1.000
Stress = 2008 g./sq, era. 24.7 40.3 45.7 60.0 74.0 120. 146.
-
-
-
3. Temp = --20°C. --2.000 - 1.778 -
1 . 0 0 0
-0.699 0.301 0.000 0.699 1.000 -
4. Temp = - 5 0 ° c . --2.000 -- 1.000 --0.699 -- 0.301 0.000 0.699 1.000 Abridged table. GR-S had no permanent set,
0.924 0.712 0.634 0.516 0.425 0.215 0.130
Stress = 5300 g./sq, c a . 35.2 37.1 46.1 52.5 63.4 74.8 118.0 150.7
1.067 0.985 0.913 0.715 0.609
Stress = 9470 g./sq, c a . 8.0 24.2 31.3 42.7 48.7 62.7 68.7
2.190 1.712 1.599 1.461 1.408 1.297 1.255
1.241 1.218 1 . 1 2 3
TIME
AND
TEMPERATURE TABLE
259
STUDIES
III
Polyisobutylene and GR-S Time Correction Factors Log K, polyisobu~ylene Constant E Viscosity (20.1 keal.) data 0.708 0.541 0.000 0.000 -- 0.783 -- 0.628 -- 1.649 -- 1.361 --2.625 --2.234 -- 3.720 -- 3.271 -- 4.960 -- 4.514
Temp. °C. 40 25 10 -- 5 -- 2 0 -- 35 -- 5 0
TABLE
Log K, GR-S Constant E Shifting of (20.0 keal.) creep curves 0.705 0.592 0.000 0.000 -- 0.779 -- 0.698 -- 1.640 -- 1.468 --2.610 --2.578 -- 3.700 -- 3.934 -- 4.940 -- 5.344
IV
Polyisobutylene: Apparent Activation Energy at Constant Reduced Modulus L o g (To/T)G E (kcal./mole)
1.400 19.8
1.191 19.8
0.976 20.2
TABLE
0,900 22.9
0.792 21.1
0.656 16.6
V
GR-S: Apparent Activation Energy at Constant Reduced Modulus (To/T)G E (kcal./mole)
Log
1.24 28:0
0.99 25.3
0.80 19.6
0.61 17.7
35.
0.47 16.7
\
\ I.C
I
-5 ° o
~.~
j
25 °
06
c
F I G , 1. P o l y i s o b u t y l e n e
I
creep curves,
0.34 16.4
0.13 15.5
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t.2
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0.~
0 . 6
0.4 -5
-4
-5 -2 -I 0 LOG REDUCED TIME (MINUTES)
I
2
FIG. 4. P o l y i s o b u t y l e n e " m a s t e r c u r v e . " T i m e r e d u c t i o n calculated f r o m steady-flow viscosities.
LOG TIME (MINUTES)
FIG. 5. G R - S creep curves.
~6~
(~. A. DA~LQI]~ST AN~D 1VI. R. t~ATF]::EL~
good agreement in the longer time region but diverge somewhat at times shorter than 10-4 min. Actual time correction values for the two methods are not equal at any point along th e curves except, of course, at the reference temperature where the correction is zero. The methods of calculation are not equivalent because the method based on steady-flow viscosity involves a continuous change in viscous flow activation energy. Brown and Tobolsky noted a change in apparent activation energy taking place at temperatures around "-45°C. for a high-molecular-weight polyisobutylene. No such variation was observed in this investigation but it might have become apparent had the measurements been extended to lower temperatures.
I.O
o LOG TI(MIMNE~ -i.ol
/ -2.0[
32
3.6 I 4.0 7.~o3
4.4
FIG. 6. GR-S: log time vs. reciprocal of temperature at constant reduced modulus. In contrast to polyisobutylene, GR-S rubber does not display a constant apparent activation energy as determined by the slopes of the log time vs. reciprocal temperature curves (Fig. 6). Nevertheless, a master curve based on a constant average activation energy (Fig. 7) shows a reasonably good superposition of data, though not as good as that obtained by shifting the reduced modulus-log time curves along the log time axis (Fig. 8). Time factors calculated from the shifting process are
TIME AND T E M P E R A T U R E STUDIES
263
2.0 1.6__ 1.2
LOG~G 0.8 0.4 0 -0~4 -7
-6
-5 -4 -3 -2 -I O LOG REDUCED TIME (MINUTES)
I
FIG. 7. G R - S " m a s t e r c u r v e . " T i m e r e d u c t i o n b a s e d on c o n s t a n t a c t i v a t i o n energy.
plotted against reciprocal temperature in Fig. 9. The activation energy appears to change abruptly with temperature at -10°C. If the apparent activation energies obtained from the log timereciprocal temperature curves are plotted against reduced modulus, it is seen that the activation energy increases in a roughly linear manner
L
__
2.0 1.6 1.2 LOG -~G 0,8
0.~ 0 (14 -8
-7
-6
-5 -4 -3 -2 -I LOG REDUCED TIME(MINUTES)
0
FIG. 8. G R - S " m a s t e r c u r v e " o b t a i n e d by shifting creep curves along t i m e axis.
C. A. DAIILQUIST AND M. R. HATFIELD
264
4
_
_
S
3 LOG k 2 I
--
i
0 I
[ i
-2 3.0
3.2
~.~
~.~
~ ,~
3.8 -.,o3
f ~.~
~.2
FIG. 9. Temperature dependence of creep curve translation factors (GR-S).
6.4 6.¢
o
5.6
o
I
5.2
2 . ~ - ~ XlO"3 •
4.8
4.4
4.0
3.6
3.27 0
4
8
To
T
12
16
G
Fro. 10. Variation in apparent activation energy with reduced modulus.
20
46
TIME AND TEMPERATURE STUDIES
I.+ 8-
~
265
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L2
_
.
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-7
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-4
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-2
.
.
-1 "
. 0
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2
LOG REDUCED TIME{MINUTES)
FIG. 11. GR-S "master curve." Time reduction based on modulus-dependentactivation energy. with increasing modulus (Fig. 10). From the least squares method E the equation representing this dependence was calculated as 2.303R
This implies that the activation energy for the elastic deformation is dependent upon the particular process, i.e., processes characterized by short retardation times have higher activation energies than those of long retardation times. This is not necessarily justified from a theoretical point of view, but modulus-time data reduced on this basis fall very nicely on a master curve with practically no scatter (see Fig. 11). This I
I
I
2.0 _ 1.6
I
I
~'k~
=
I
I
I
I +
I
I
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\ \
+ - E vA.,Es W,T. REOUCEO0 --
1.2_ LOG'~rG 0.8_
OA--
0~ -0
-IO
I
-9
I
-8
I
-7
I
-6
I
-5
I
-4
I
-3
f
-2
I
-I
LOG REDUCED TIME (MINUTES) Fro. 12. Comparison of GR-S master curves.
I
0
I
I
266
C . A . DAHLQUIST AND lYI. R. ttATFIELD
curve coincides with the other two in the long time region from 10 2 to 10 -5 rain., b u t at times shorter t h a n 10 -2 rain., log modulus increases at a m u c h slower rate (Fig. 12). T h e portion of the curve represented b y the broken line is based on extrapolated activation energies. ACKNOWLEDGMENTS
The authors wish to express their appreciation to R. S. Marvin for supplying the standard polyisobutylene, and to W. E. Luedke for conducting the experimental nleasurements.
SUMMARY The constant-stress method recently described by Dahlquist et al. is well-suited fo~ the study of the time and temperature dependence of viscoelastic properties. Creep data are given for polyisobutylene and GR-S rubber in the time range of 0.01 to i0 min. and the temperature range of 40 to -50°C. The data were transferred into master curves of log reduced modulus vs. log reduced time. The polyisobutylene data superposed satisfactorily using either (a) a time reduction method based on a constant apparent activation energy for elastic deformation, or (b) a time reduction method based on steadyflow viscosities. The two master curves coincide except in the short time range. In contrast to polyisobutylene, GR-S showed a considerable variation in activation energy with modulus. Three master curves are presented, the first calculated on the basis of a constant average activation energy, the second obtained by shifting individual modulus curves along the log time axis, and the third derived by assuming a linear dependence of activation energy on modulus. The scattering of the data decreases in order from the first to the third. The first and second curves coincide completely within experimental error. The third curve coincides with the first and second in the time range of 10 -2 to 101.5 rain. but diverges considerably at time intervals less than 10 -2 min. As yet the concept of a modulus-dependent activation energy must be considered of doubtful theoretical significance. REFERENCES 1. ANDRADE~E. I~. DA C., Proc. Roy. Soc. (London) 84, 1 (1910). 2. ANDREWS,R. D., AND TOBOLSKY, A. V., J. Chem. Phys. 13, 3 (1945). 3. ANDREWS, R. D., I-IoFMAN-BANG,l~., AND TOBOLSKY~ A. V., J. Polymer Sci. 3, 669 (1948). 4. BROWN, G. M., AND TOBOLSKY, A. V., J. Polymer Sci. 6, 165 (1951). 5. DAHLQUIST, C: A., I~IENDRICKS,J. O., AND TAYLOR, I~. W., Ind. Eng. Chem. 43, 1404 (1951).
TIME AND TEMPERATURE STUDIES
267
6. FERRY, J. D., J. Am. Chem. Soc. 72, 3746 (1950). 7. Fox, T. G., JR., AND FLORY, P. J., J. Am. Chem. Soe. 70, 2384 (1948). 8. MARVnV, R. S., Interim Report on Coop. Program on Dynamic Testing. Natl. Bureau Standards (U. S.), April 25, 1951. 9. SCOTT,]%. L., CARTER,W. C., AND MAG•T, ~Vi., J. Am. Chem. Soc. 71, 220 (1949). 10. VIEWEO,VON R., AlVDSCH~]~I~ER, W., Kunststoffe 31, 215 (1941). 11. WORK, R. N., Tech. Rept. to Office Naval Research, Pro]. No. 0NR(QMC), NR-033-314, October 30, 1950.
INTRODUCTION The effect of time and temperature on the viscoelastic behavior of rubbery materials is of great practical significance and has long been the subject of extensive investigation and theoretical interpretation. It has been shown that time and temperature have an equivalent effect on the elastic modulus of linear rubbery polymers (2). A quantitative relationship between time and temperature can be derived from the apparent activation energy for viscous flow (6) or for elastic deformation (3). The temperature dependence of the modulus can then be reduced to an equivalent time dependence, and data covering a wide range of time and temperature can be represented by a single "master curve" of reduced variables. The constant-stress method (5) used in this laboratory for characterization of soft polymers is well-suited for study of the time and temperature dependence of viscoelastic properties. The data are similar in many respects to the stress relaxation data of Andrews et al. (2,3,4). It was of interest to investigate the treatment of creep data by reduced variables and to compare the two methods cited above. It was also of interest to extend the treatment to polymers other than polyisobutylene, preferably to one which can be subjected to high elongation without undergoing crystallization. THEORY The modulus of elasticity, as used in this paper, is defined as the stress per unit elongation. A discussion of the applicability of such a property to constant-stress measurements on soft polymers has been given in an earlier paper (5). Reduced modulus is calculated by multiplying the experimental value by the factor 298/T, where T is the absolute temperature. This factor reduces all modulus contributions associated with entropy change to the Presented at the Annual Meeting of the Society of Rheology, Chicago, Illinois, October 24-27, 1951. 2Contribution No. 39. 253
~54
C. A. DAItLQUIST AND M. R. ItATFIELD
same reference temperature, 298°K., as predicted by the kinetic theory of rubberlike elasticity. If all retardation processes contributing to the total creep have the same temperature dependence, they c a n be characterized by a single activation energy. This activation energy can be calculated by use of the Arrhenius equation in which the reciprocal of the reaction rate at temperature T is replaced by the time, t, required to attain a given modulus. Thus t = A l e EIRT
El-1
where A t is a constant, E the activation energy, and R the gas constant. A plot of the natural logarithm of time versus the reciprocal temperature yields a straight line of slope E / R . The correction factor, K, which reduces the time corresponding to a given modulus at one temperature to that corresponding to the same modulus a t another temperature, is given by Eq. [-11 written for two temperatures K -
to t
_
e E/RT°
e EIRT"
~2~
A master curve of reduced modulus versus reduced time can be obtained by use of Eq. [21. (Andrews, Hofman-Bang, and Tobolsky in Ref. (3) give a more detailed discussion of the theory presented here.) Creep curves may be shifted along the log time axis to form a single master curve which, if the activation energy is constant, will coincide with that described above. The temperature dependence Of the time correction factor, as calculated from the actual shift, is given by the Arrhenius equation K = Ae -zIRT. [31 It should be noted that Eq. [-31, which gives the temperature dependence of K, becomes identical with Eq. E2~ if A is evaluated for the reference temperature at which K is equal to unity. A treatment based on steady-flow viscosity allows a similar reduction of time to arrive at a master curve of reduced modulus and reduced time. Recently, Ferry (6) derived the relationships which apply to dynamic properties, and the treatment for creep measurements is quite similar to that presented by Ferry. The polymeric substance is assumed to behave like a group of Voigt units in which each elasticity mechanism is represented by a modulus G~ and each flow process by riGi. Under the restriction of constant stress, the elongation at time t is given by the well-known relationship ~ = SE~(IlG~)
(1 - e-ricO,
E4~
TIME AND
TEMPERATURE
STUDIES
~
w h e r e S is t h e a p p l i e d stress. I f m o d u l u s is defined as stress p e r u n i t s t r a i n , E q . [-4] b e c o m e s 1/G, = E~(1/ai)
(1
-
e-u'i).
[5]
T h e t e m p e r a t u r e d e p e n d e n c e of all r e t a r d a t i o n m e c h a n i s m s is a s s u m e d t o b e g i v e n b y a single f a c t o r A f d e p e n d e n t o n l y u p o n t h e p o l y m e r m a t e r i a l a n d t e m p e r a t u r e . T h e m o d u l u s G~ of r u b b e r l i k e s p r i n g s is p r o p o r t i o n a l to t h e a b s o l u t e t e m p e r a t u r e so t h a t G~ = ( T / T o ) G . ,
where the subscript zero designates a reference temperature. F5] at temperature T is then
1/G,
= E~(To/T)
(1/Gi0) (1 - e-'~T~O.
[6] Equation [7]
I t follows t h a t Gt = ( T / T o ) G o u a r .
[8]
F e r r y (6) h a s s h o w n t h a t t h e s t e a d y - f l o w v i s c o s i t y n a t t e m p e r a t u r e T is given by = A T pT~?o poTo'
[9]
w h e r e p is t h e d e n s i t y . M a r v i n (8) h a s c a l c u l a t e d A T v a l u e s for p o l y i s o b u t y l e n e f r o m p r e c i s e v i s c o s i t y (7) a n d l i n e a r e x p a n s i o n (11) d a t a . T h e s e d a t a were u s e d in t h i s paper. EXPERIMENTAL The constant-stress apparatus was recently described by Dahlquist, Hendricks, and Taylor (5). A strip of polymer cut from a thin sheet or film is loaded with a weight which sinks into a liquid as the film stretches. By suitable design of the weight, which incidentally, takes the form of a hyperboloid, the load can be reduced in proportion to the change in cross-sectional area of the stretching film. The same technique has been used by Andrade (1) to study viscous flow in metallic wires. The apparatus has recently been modified to the extent that the thread release has been replaced by a magnetic release. The technique of supporting the tension weight by a thread and subsequently transferring the load to the test strip by burning the thread proved to be rather cumbersome, and a release mechanism was developed wherein the load is initially supported by an electromagnet. A soft iron rod, 1 ~ in. long X ~ in. diameter, inserted in the tab attached to the lower end of the vertically mounted test strip serves as an armature for the magnet, and supports the hyperbolic weight which is suspended by a fine wire. The load is transferred to the test strip when the armature is released by breaking the field circuit of the magnet. The tab attached to the upper end of the test strip is held in a movable clamp to permit taking up of slack in the specimen after it has been positioned in the tester. Temperature control was accomplished by enclosing the test specimen in a chamber, 5 X 4 X 14 in., with an opening at the top leading into a Dry Ice container. Air was forced through the Dry Ice and down into the test chamber, an inverted funnel just below the cold air inlet serving as a baffle to break up the air stream and distribute it uni-
256
c. A. DAHLQUIST AND M. R. HATFIELD
formly throughout the chamber. The temperature was adjusted by regulating the volume of incoming air. Temperatures above 25°C. were obtained by replacing the Dry Ice container with a preheating unit. The test strips were cut from sheets approximately 0.020 in. thick cast h'om solution on amalgamated tin panels. Polyisobutylene was dissolved in n-heptane without preliminary treatment, while GI~-S rubber received a brief cold milling. The drying schedule was as follows: (a) 24 hr. slow drying under cover, (b) 24 hr. uncovered, and (c) 16 hr. vacuum drying at 50°C. Further details of the film-casting technique are described in the earlier publication (5). The stress conditions were controlled by regulating the initial cross-sectional area of the test specimen. Had the same stress been used in all of the experiments, the deformations would have been very small at the lowest temperatures, or excessively large, leading to rupture of the specimens, at the highest temperatures. To avoid this an attempt was made to control the stress to give approximately the same 10-rain. elongation at all temperatures, ttowever, it was difficult to prepare specimens of sufficiently small cross section to fulfill the stress requirements for the -20, -35, and -50°C. tests. Since polyisobutylene tends to crystallize when extended beyond 150%, it s deformations were held below 100%. The deformation of GR-S, which does not crystallize, was permitted to exceed 200% in only one instance. At the higher temperatures, polyisobutylene exhibited a small degree of permanent set. This was subtracted from the total deformation to give the purely elastic deformation, and the modulus values were based upon the elastic component. GR-S showed no perma.hent set in these experiments. The modulus values were calculated according to the conventional "stress over strain" formula: stress × initial length G= deformation The polyisobutylene (Vistanex) was furnished by R. S. Marvin of the National Bureau of Standards.It had been carefully selected as a reference standard for comparative dynamic testing by a number of laboratories and has been described very completely by Marvin (8). It has a viscosity average molecular weight of 1.35 X 10~, a density3 of 0.913 g. cc. at 25°C., and a coefficient of linear expansion of 0.58 X 10-3/°C. GR-S X-274 was taken from a bale of commercial gum rubber. It was characterized as follows: intrinsic viscosity in benzene, 2.41; viscosity average molecular weight, 306,000 (9); Mooney viscosity, 66; gel content, zero. The measured density was 0.935. To determine the change in density with temperature the linear coefficient of expansion was taken as 1.85 X 10-4/°C., the value reported by Vieweg and Schneider (10) for Buna S-I. t~ESULTS AND DISCUSSION A b r i d g e d e x p e r i m e n t a l d a t a are p r e s e n t e d i n T a b l e s I a n d I I . Fig. i a n d 5, w h i c h c o n t a i n all of t h e d a t a , show plots of " r e d u c e d m o d u l u s " as a f u n c t i o n of t h e l o g a r i t h m of t i m e a t t h e v a r i o u s t e m p e r a t u r e s considered. T h e " r e d u c e d m o d u l u s " was d e t e r m i n e d b y m u l t i p l y i n g t h e e x p e r i m e n t a l m o d u l u s b y t h e f a c t o r 2 9 8 / T , where T is t h e a b s o l u t e t e m p e r a t u r e . If all r e t a r d a t i o n processes h a v e t h e s a m e temperat~ure dep e n d e n c e , a m a s t e r c u r v e of r e d u c e d m o d u l u s vs. t i m e m a y b e o b t a i n e d 3 Private communication from Dr. Marvin.
TIME AND TEMPERATURE STUDIES
257
b y m e r e l y s h i f t i n g t h e c r e e p c u r v e s a l o n g t h e l o g t i m e axis. T h i s i m p l i e s t h a t all t i m e v a l u e s a t a g i v e n t e m p e r a t u r e are multiplied by a single c o r r e c t i o n f a c t o r . A n d r e w s et al. (3) h a v e s h o w n t h a t t h i s c o r r e c t i o n f a c t o r is r e l a t e d t o a n a p p a r e n t a c t i v a t i o n e n e r g y . A c t i v a t i o n e n e r g y v a l u e s w e r e c a l c u l a t e d f r o m a p l o t of l o g t i m e a t constant reduced modulus vs. reciprocal temperature. Figs. 2 and 6 TABLE I Polyisobutylene Creep Data ~
Total Viscous Log thne elongation flow min. % % 1. Temp = 40°C. Stress = 2200 g./sq, cm. - 1.778 31.5 --0.699 38.6 -0.301 43.6 -0.000 49.4 0.3 0.699 70.2 2.7 1.000 86.5 7.3 - -
Elas¢.ic elonga%ion %
Log (To/T)G (G in megadynes/sq, cm.)
31.5 38.6 43.6 49.1 67.5 79.2
' 0.824 0.726 0.673 0.621 0.483 0.413
2. Temp = 10°C. -- 1.778 0.699 0.301 0.000 0.699 1.000
Stress = 3710 g./sq, cm. 43.6 -51.7 -56.0 -60.5 -75.4 0.3 85.3 0.7
43.9 52.0 56.4 60.8 75.1 84.6
0.943 0.869 0.835 0.802 0.708 0.657
3. Temp = - 2 0 ° C . -- 1.778 --1.177 --0.699 0.301 0.000 0.699 1.000
Stress = 4240 g./sq, cm. 19.2 -34.1 -41.2 -45.9 -49.0 -58.4 -61.5 --
19.2 34.1 41.2 45.9 49.0 58.4 61.5
1.408 1.158 1.075 1.029 0.999 0.924 0.900
4. Temp = - 5 0 ° C . 0.301 0:699 1.000 a Abridged table.
Stress = 7190 g./sq, cm. 24.5 -44.7 -60.3 --
24.5 44.7 60.3
1.581 1.324 1.193
- -
- -
- -
represent these plots for the two polymers under investigation. The activ a t i o n e n e r g y d e r i v e d f r o m t h e p o l y i s o b u t y l e n e d a t a (see T a b l e I V ) a p pears to be reasonably constant over the entire range, and the average v a l u e of 20.1 k c a l . is i n s u b s t a n t i a l a g r e e m e n t w i t h t h e v a l u e 19.5 o b tained by Brown and Tobolsky from stress relaxation measurements in the temperature range 0 to -45°C. (4).
258
C.A.
DAHLQUIST AND M. R. ttATFIELD
A master curve for polyisobutylene based on a constant activation e n e r g y of 20.1 k c a l . is s h o w n i n F i g . 3. T h e t r a n s l a t e d d a t a s u p e r p o s e nicely except for some scattering at the very low temperatures (short reduced time). A s e c o n d m a s t e r c u r v e b a s e d o n s t e a d y - f l o w v i s c o s i t i e s is p r e s e n t e d i n F i g . 4. T h e d a t a s h o w m o r e s c a t t e r t h a n i n F i g : 3. T h e t w o c u r v e s a r e i n TABLE II GR-S X274 Creep Data a Log time min.
Total elongationb
%
Log (To/T)G (G in megadynes/sq, cra.)
1. Temp = 40°C. --2.000 - 1:778 1.000 0.699 --0.30l 0.00O 0.699 1.000
Stress = 1306 g./sq, c a . 29.6 0.615 34.1 0.554 52.5 0.365 63.5 0.283 81.8 0.176 100.7 0.083 166.0 -0.136 208.5 --0.233
2. Temp = 10°C. 2.000 - 1.000 - 0.699 -0.301 0.000 0.699 1.000
Stress = 2008 g./sq, era. 24.7 40.3 45.7 60.0 74.0 120. 146.
-
-
-
3. Temp = --20°C. --2.000 - 1.778 -
1 . 0 0 0
-0.699 0.301 0.000 0.699 1.000 -
4. Temp = - 5 0 ° c . --2.000 -- 1.000 --0.699 -- 0.301 0.000 0.699 1.000 Abridged table. GR-S had no permanent set,
0.924 0.712 0.634 0.516 0.425 0.215 0.130
Stress = 5300 g./sq, c a . 35.2 37.1 46.1 52.5 63.4 74.8 118.0 150.7
1.067 0.985 0.913 0.715 0.609
Stress = 9470 g./sq, c a . 8.0 24.2 31.3 42.7 48.7 62.7 68.7
2.190 1.712 1.599 1.461 1.408 1.297 1.255
1.241 1.218 1 . 1 2 3
TIME
AND
TEMPERATURE TABLE
259
STUDIES
III
Polyisobutylene and GR-S Time Correction Factors Log K, polyisobu~ylene Constant E Viscosity (20.1 keal.) data 0.708 0.541 0.000 0.000 -- 0.783 -- 0.628 -- 1.649 -- 1.361 --2.625 --2.234 -- 3.720 -- 3.271 -- 4.960 -- 4.514
Temp. °C. 40 25 10 -- 5 -- 2 0 -- 35 -- 5 0
TABLE
Log K, GR-S Constant E Shifting of (20.0 keal.) creep curves 0.705 0.592 0.000 0.000 -- 0.779 -- 0.698 -- 1.640 -- 1.468 --2.610 --2.578 -- 3.700 -- 3.934 -- 4.940 -- 5.344
IV
Polyisobutylene: Apparent Activation Energy at Constant Reduced Modulus L o g (To/T)G E (kcal./mole)
1.400 19.8
1.191 19.8
0.976 20.2
TABLE
0,900 22.9
0.792 21.1
0.656 16.6
V
GR-S: Apparent Activation Energy at Constant Reduced Modulus (To/T)G E (kcal./mole)
Log
1.24 28:0
0.99 25.3
0.80 19.6
0.61 17.7
35.
0.47 16.7
\
\ I.C
I
-5 ° o
~.~
j
25 °
06
c
F I G , 1. P o l y i s o b u t y l e n e
I
creep curves,
0.34 16.4
0.13 15.5
©
©
m
p
0
,3
&
/
I
/
Q
0
/
/
s
0
r
J
O..q.
~p
©
©
b~
~o~
o
~o "Qs>
.
~'~
~o
~~~~-~
_
!
[
"
~
d
b~
TIME
1.6
i
AND
TEMPER~4.TURE
-!
STUDIES
I
! 1
]
i
I
i
1.4
261
!
t.2
LOG-~G I.C__
0.~
0 . 6
0.4 -5
-4
-5 -2 -I 0 LOG REDUCED TIME (MINUTES)
I
2
FIG. 4. P o l y i s o b u t y l e n e " m a s t e r c u r v e . " T i m e r e d u c t i o n calculated f r o m steady-flow viscosities.
LOG TIME (MINUTES)
FIG. 5. G R - S creep curves.
~6~
(~. A. DA~LQI]~ST AN~D 1VI. R. t~ATF]::EL~
good agreement in the longer time region but diverge somewhat at times shorter than 10-4 min. Actual time correction values for the two methods are not equal at any point along th e curves except, of course, at the reference temperature where the correction is zero. The methods of calculation are not equivalent because the method based on steady-flow viscosity involves a continuous change in viscous flow activation energy. Brown and Tobolsky noted a change in apparent activation energy taking place at temperatures around "-45°C. for a high-molecular-weight polyisobutylene. No such variation was observed in this investigation but it might have become apparent had the measurements been extended to lower temperatures.
I.O
o LOG TI(MIMNE~ -i.ol
/ -2.0[
32
3.6 I 4.0 7.~o3
4.4
FIG. 6. GR-S: log time vs. reciprocal of temperature at constant reduced modulus. In contrast to polyisobutylene, GR-S rubber does not display a constant apparent activation energy as determined by the slopes of the log time vs. reciprocal temperature curves (Fig. 6). Nevertheless, a master curve based on a constant average activation energy (Fig. 7) shows a reasonably good superposition of data, though not as good as that obtained by shifting the reduced modulus-log time curves along the log time axis (Fig. 8). Time factors calculated from the shifting process are
TIME AND T E M P E R A T U R E STUDIES
263
2.0 1.6__ 1.2
LOG~G 0.8 0.4 0 -0~4 -7
-6
-5 -4 -3 -2 -I O LOG REDUCED TIME (MINUTES)
I
FIG. 7. G R - S " m a s t e r c u r v e . " T i m e r e d u c t i o n b a s e d on c o n s t a n t a c t i v a t i o n energy.
plotted against reciprocal temperature in Fig. 9. The activation energy appears to change abruptly with temperature at -10°C. If the apparent activation energies obtained from the log timereciprocal temperature curves are plotted against reduced modulus, it is seen that the activation energy increases in a roughly linear manner
L
__
2.0 1.6 1.2 LOG -~G 0,8
0.~ 0 (14 -8
-7
-6
-5 -4 -3 -2 -I LOG REDUCED TIME(MINUTES)
0
FIG. 8. G R - S " m a s t e r c u r v e " o b t a i n e d by shifting creep curves along t i m e axis.
C. A. DAIILQUIST AND M. R. HATFIELD
264
4
_
_
S
3 LOG k 2 I
--
i
0 I
[ i
-2 3.0
3.2
~.~
~.~
~ ,~
3.8 -.,o3
f ~.~
~.2
FIG. 9. Temperature dependence of creep curve translation factors (GR-S).
6.4 6.¢
o
5.6
o
I
5.2
2 . ~ - ~ XlO"3 •
4.8
4.4
4.0
3.6
3.27 0
4
8
To
T
12
16
G
Fro. 10. Variation in apparent activation energy with reduced modulus.
20
46
TIME AND TEMPERATURE STUDIES
I.+ 8-
~
265
'
L2
_
.
0.E LOGT'?G 0.4
~0.4
.
-I0
-9
+8
-7
-6
~5
-4
-3
-2
.
.
-1 "
. 0
i I
2
LOG REDUCED TIME{MINUTES)
FIG. 11. GR-S "master curve." Time reduction based on modulus-dependentactivation energy. with increasing modulus (Fig. 10). From the least squares method E the equation representing this dependence was calculated as 2.303R
This implies that the activation energy for the elastic deformation is dependent upon the particular process, i.e., processes characterized by short retardation times have higher activation energies than those of long retardation times. This is not necessarily justified from a theoretical point of view, but modulus-time data reduced on this basis fall very nicely on a master curve with practically no scatter (see Fig. 11). This I
I
I
2.0 _ 1.6
I
I
~'k~
=
I
I
I
I +
I
I
I--CONSTANT E _ 2--TRA"S'ATED CREEP CURVES
\ \
+ - E vA.,Es W,T. REOUCEO0 --
1.2_ LOG'~rG 0.8_
OA--
0~ -0
-IO
I
-9
I
-8
I
-7
I
-6
I
-5
I
-4
I
-3
f
-2
I
-I
LOG REDUCED TIME (MINUTES) Fro. 12. Comparison of GR-S master curves.
I
0
I
I
266
C . A . DAHLQUIST AND lYI. R. ttATFIELD
curve coincides with the other two in the long time region from 10 2 to 10 -5 rain., b u t at times shorter t h a n 10 -2 rain., log modulus increases at a m u c h slower rate (Fig. 12). T h e portion of the curve represented b y the broken line is based on extrapolated activation energies. ACKNOWLEDGMENTS
The authors wish to express their appreciation to R. S. Marvin for supplying the standard polyisobutylene, and to W. E. Luedke for conducting the experimental nleasurements.
SUMMARY The constant-stress method recently described by Dahlquist et al. is well-suited fo~ the study of the time and temperature dependence of viscoelastic properties. Creep data are given for polyisobutylene and GR-S rubber in the time range of 0.01 to i0 min. and the temperature range of 40 to -50°C. The data were transferred into master curves of log reduced modulus vs. log reduced time. The polyisobutylene data superposed satisfactorily using either (a) a time reduction method based on a constant apparent activation energy for elastic deformation, or (b) a time reduction method based on steadyflow viscosities. The two master curves coincide except in the short time range. In contrast to polyisobutylene, GR-S showed a considerable variation in activation energy with modulus. Three master curves are presented, the first calculated on the basis of a constant average activation energy, the second obtained by shifting individual modulus curves along the log time axis, and the third derived by assuming a linear dependence of activation energy on modulus. The scattering of the data decreases in order from the first to the third. The first and second curves coincide completely within experimental error. The third curve coincides with the first and second in the time range of 10 -2 to 101.5 rain. but diverges considerably at time intervals less than 10 -2 min. As yet the concept of a modulus-dependent activation energy must be considered of doubtful theoretical significance. REFERENCES 1. ANDRADE~E. I~. DA C., Proc. Roy. Soc. (London) 84, 1 (1910). 2. ANDREWS,R. D., AND TOBOLSKY, A. V., J. Chem. Phys. 13, 3 (1945). 3. ANDREWS, R. D., I-IoFMAN-BANG,l~., AND TOBOLSKY~ A. V., J. Polymer Sci. 3, 669 (1948). 4. BROWN, G. M., AND TOBOLSKY, A. V., J. Polymer Sci. 6, 165 (1951). 5. DAHLQUIST, C: A., I~IENDRICKS,J. O., AND TAYLOR, I~. W., Ind. Eng. Chem. 43, 1404 (1951).
TIME AND TEMPERATURE STUDIES
267
6. FERRY, J. D., J. Am. Chem. Soc. 72, 3746 (1950). 7. Fox, T. G., JR., AND FLORY, P. J., J. Am. Chem. Soe. 70, 2384 (1948). 8. MARVnV, R. S., Interim Report on Coop. Program on Dynamic Testing. Natl. Bureau Standards (U. S.), April 25, 1951. 9. SCOTT,]%. L., CARTER,W. C., AND MAG•T, ~Vi., J. Am. Chem. Soc. 71, 220 (1949). 10. VIEWEO,VON R., AlVDSCH~]~I~ER, W., Kunststoffe 31, 215 (1941). 11. WORK, R. N., Tech. Rept. to Office Naval Research, Pro]. No. 0NR(QMC), NR-033-314, October 30, 1950.