Temperature Dependence of Dynamic Properties of Elastomers; Relaxation Distributions JOHN D. FERRY, EDWIN R. FITZGERALD, LESTER D. GRANDINE, JR., AND MALCOLM L. WILLIAMS University of Wisconsin, Madison, Wis.
T
HE
B y the use of reduced variables, the temperature denot require any analytical experimental pendence and frequency dependence of dynamic mechaniexpression for either the measurement of dynamic mechanical properties ‘cal properties of rubberlike materials can be interrelated temperature dependence or of a rubberlike material without any arbitrary assumptions about the functional the frequency dependence, in small sinusoidal deforform of either. The definitions of the reduced variables though it does involve some are based on some simple assumptions regarding the nasimple assumptions about mations alwitys consists of ture of relaxation processes. The real part of the reduced determining two independthe nature of elastic and ent quantities a t each fredynamic rigidity, plotted against the reduced frequency, relaxation mechanisms; gives a single composite curve for data over wide ranges of these are explicitly stated. quency. These two quantifrequency and temperature; this is true also for the imties can be expressed in variTHE METHOD OF REDUCED ous ways: here, as in the aginary part of the rigidity or the dynamic viscosity. The VARIABLES paper by Marvin ( I I ) , the real and imaginary parts of the rigidity, although independent measurements, are interrelated through the disWe assume that elastic dynamic properties are tribution function of relaxation times, and this relation energy is stored in the defdescribed by G‘ and G“, the provides a check on experimental results. First and secormation of a rubberlike real and imaginary parts of ond approximation methods of calculating the distribumaterial (whether vulcanthe dynamic shear modulus, or by G’ and v’, where q‘ is ized or not) by a variety of tion function from dynamic data are given. The use of the distribution function to predict various types of timemechanisms of the configuthe real part of the complex rational or entropy-decrease visrosity(q’ = C ” / w , wherew dependent mechanical behavior is illustrated. is the circular frequency). type, each of which makes The relations between these a contribution Gi t o the and other properties such as damping and energy dissipation have instantaneous rigidity. The stress on the ith mechanism decays been given by Marvin. exponentially with a relaxation time, T ( . This scheme is equivThis paper is concerned with two procedures for treating such alent to the “generalized Maxwell model” ( I ; 8, IO). The dynamic properties and the steady flow viscosity, 7, are then given experimental data to facilitate interpretation and practical use. The fir& is a method for combining the frequency and temperby the following equations (1, IO): ature dependence of G’ and G” by introducing certain reduced GI(@) = ZGiW2T,2/(1 4-w 2 7 f ) (la) variables. The second is a method for combining G’ and G” (or G’ and q ’ ) and checking one against the other by the distribu~ “ ’ ( c o= ) x G ~ ~ i / ( l w%:) (Ib) tion function of relaxation times. Once this function is obtained, ~ ’ ( w )= ZGiri/(1 w2r?) (IC) it is possible in principle to predict any sort of time-dependent
+ +
behavior under small stresses.
q = Zff1Ti
(Id)
It is now assumed that each G‘$ is proportional to the absolute temperature, T (since ,it involves configurational or rubberlike It haa long been recopized that there is a mnnection between elasticity), and to the density, p (since the mechanical quantities temperature and. frequency dependence of dynamic n ~ ~ h a n i c a l refer to a unit cube of material, and thermal expansion decreases properties and that a decrease in temperature is a t least qualitamechanism, within this the number of molecules, and tively equivalent to an increase in frequency. Another cube), and also that all 7 f have ~ the same temperature dependfestation of the connection is that under conditions where the freence-that is, when the temperature is changed from a reference quency dependence of G’ is slight, the temperature dependence is T~to T , every T l is multiplied by the same factor UT. It follows slight also, whererts a sharp change of G’ with frequency is always (,5,6) that the dynamic properties a t the two temperatures, T and accompanied by a sharp change with temperature. TO,are related thus Investigators have sometimes combined dynamic data a t dif(Tp/Topo)G,Xw~r) (2a) G‘(0) ferent frequencies and temperatures by plotting, for example, the equivalent of G‘ against the logarithm of the frequency and shiftG“(w) (TP/ToPo)G~)(~uT) (2b) ing the curves along the abscissa axis. However, the conditions under which such a shift would theoretically be justified have usu11’(w) = (uTTP/ToPo)11~(waT) (2c) ally not been made clear. Sometimes the temperaturesand fre11 = (mTP/ToPo)rlO (2d) quency (or time) dependence of mechanical behavior have been combined in an explicit algebraic formula (5), but it is unlikely where the quantities without subscript refer to temperature T that this can be successful for dynamic properties over wide and those with subscript 0 refer to the reference temperature, TO. ranges of either variable, since experience indicates that G’ and Based on these equations, it is easy to define reduced variables G“ vary in too complicated a manner to fit a simple empirical which will be independent of temperature if our assumptions are equation. The method of reduced variables outlined here does valid: TEMPERATURE DEPENDENCE OF DYNAMIC PROPERTIES
=i
April 1952
INDUSXRIAL AND ENGINEERING CHEMISTRY
703
-ELASTOMERS-Dynamic
PropertiesThe values of U T are plotted logarithmically against the absolute temperature in Figure 3. The plot is not a straight line, so the temperature dependence of a T is not a simple exponential function. For an unvulcanized rubber, like polyisobutylene, the temperature dependence of a T is related to that of the steady flow viscosity; from Equation 2d, ~ T / V O= aTTp/Topo. This relation has in fact been found to hold within experimental error for polyisobutylene (6, l a ) , and it provides an alternative method for obtaining the reduction factor aT. Composite curves such as those of Figure 2 may be used to predict dynamic properties outside the temperature and frequency ranges of measurement. A further generalization and a check of one curve against the other is provided by the distribution function of relaxation times,
8.0
4.O
7.5
3.5
e
W
-
0,
RELAXATION DISTRIBUTION FUNCTION
CI,
0
0
7,o
-
3.O
6.5
\ 2.5 2
+
4
3
It is now generally accepted that the time-dependent mechanical behavior of a rubberlike material cannot be described in terms of a small number of relaxation times, corresponding to a mechanical model of a small number of springs and dashpots; the most useful description is by a continuous distribution of relaxation times, corresponding to a mechanical model of infinite extent. From such a distribution function can be calculated, in principle, the behavior under any kind of time-dependent stresses of Emall magnitude, including the sinusoidally varying stresses used in most dynamic measurements as well as more complicated loading patterns which may be of practical interest. The distribution function employed here is G d In 7 , which represents the differential contribution to rigidity with relaxation times lying between In T and In T d In T . Andrews (2) refers to a similar function for contributions to Young's modulus, which under most conditions is greater than Q by a factor of 6.91. There are exact equations expressing experimental dynamic properties in terms of a,obtained from Equations la-Id by replacing the summations by integrals:
log w Figure 1. Dynamic Rigidity (Ascending Curves) and Dynamic Viscosity (Descending Curves) of Polyisobutylene, Molecular Weight 1,200,000, at Various Temperatures, Plotted Logarithmically against Circular Frequency
4.5 7: =
q'Topo/~rTp;
wp
=
UUT
(3)
If Ga, Gi, or q l is plotted against up,a single curve should result from data a t all temperatures and frequencies. The method may be illustrated by data for a polyisobutylene sample of molecular weight 1,200,000 obtained with a new doubIe transducer apparatus which is described elsewhere (8). The original data are shown in Figure 1, which depicts families of curves for G' and ?' plotted logarithmically against log w a t different temperatures. The reduction factors a T for temperatures other than 25" C. (which is chosen as standard) must be found empirically; this is done by plotking log q'Topo/Tp against log w and shifting the curves along a line with a slope of -1 until they coincide (6); the distance moved in either the horizontal or vertical direction is log U T . The values of UT thus determined, when used to calculate the reduced variables, Gi, vi, and up,provide a single composite curve not only for q' but also for G' (Figure 2). 704
7.5
4 .O
~
-a
6
7.0
-
0,
0
G 9,
0 .,.
3 .s 6.5
3.O
I .o
2.0
3.0
4.0
5.0
log W O T Figure 2.
Data of Figure 1 Plotted with Reduced Variables
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Vol. 44, No. 4
ELASTOMERS-Dynamic
Properties-
r > > l / w , the integrand is practically equal to 9. If the distribution is broad, the last region will contribute most of the in-
0
0 d In r. J l nw Differentiation with respect t o the lower limit of this expression gives an explicit formula for 0 in terms of G'. Similar treatment of the other dynamic properties yields the following first approximation formulas (6):
tegral, which can therefore be approximated by
- 0.5
9( -In
W )
= G'(d log G'/d log
Q (-In w ) = G " ( l
I-
o rn
Q
-0
-1.5
I
(5a) (5b)
log v ' / d log w )
(5c)
w ) = -wq'(d
The value of 9 corresponding to a given T may thus be calculated from the slope of a log-log plot of any dyngmic property against frequency, measured at the value of w which is the reciprocal of T. To the extent that this first approximation is valid and the experimental data are accurate, the calculations of from ff' and from G" or 9' should agree. Values obtained from the curves of Figure 2, represedting the function as reduced to 25' C., are plotted in Figure 4 and show good agreement. I n other cases, however, discrepancies of 20y0 or greater have been observed between 0values from the two experimental sources. Experimental measurements have now progressed to the point where this is beyond experimental error, and an improved calculation is therefore desirable.
-1.0
I
3.2 IOOO/T
3.0
(-In
W )
- d log G"/d log w )
3.4
CALCULATION OF +-SECOND APPROXIMATION
It is found experimentally that
Figure 3. Logarithm, of Reduction Factor aT Plotted against Reciprocal Absolute Temperature
(44 (4b)
J--m
(40)
@ as derived by the first approximation is often a simple power function of T over a range of several powers of ten. Thus, the straight line in Figure 4 corresponds to @ = kr-O.8 from r = lo-* to r = 10-1.6 second. This region has been previously identified in polyisobutylene from earlier measurements (6). At much longer times, on the other hand, as shown by stress relaxation measurements @), 9 is flat, corresponding to hO.At still longer times, it again decreases with increasing r. To obtain a second approximation, it is assumed that within a limited region @ may be represented by the power law 0 = kr-m.
It is not so easy, however, to obtain 0 from the experimental quantities; ordinarily approximation methods are employed. A f i s t a p p r o x i m a t i o n method which has been used for the real part of the dynamic Young's modulus by several authors ( 9 , l a ) and has been extended (6) to ff', G",and T', as well as a new second approximation method, are outlined here. Another type approximation has been recently recommended by ter Haar (16,16). Unfortunately, this method is successful only when @ is inversely proportional to T, which is the case ter Haar happens to choose as an illustration; for other cases it gives highly erroneous results. CALCULATIQN OF +-FIRST APPROXIMATION
April 1952
Q
0 a, 0
- 6 0
From
G'
p
From
3'
5
The range of integration in Equation 4s may be divided into three regions: where r < < l/w, the integrand is Practically zero; where 7 the integrand is finite but less than 9; where
-
7
-I Figure 4. Logarithmic Distribution of Relaxation Times as Derived in First Approximation from Dynamic Rigidity and from Dynamic Viscosity
I N D U S T R IA L A N D E N G I N E E R I N G C H E,MI S T R Y
705
dLASTOMERS-Dynamic Table I.
Properties-
Calculation of Mechanical Behavior from @
Property
Exact Formula
Dynamic rigidity, real (G’)
Eq. 4a
Dynamic rigidity, imaginary (a”)
Eq. 4b
S--:
In d d In r
s--:”
Eq. 4c
(1’)
”
-
w
Dynamic viscosity, real
Steady flowviscosity
Approximate Formula
d d In
Elastic energy stored in steady flow ( W )
(3*,’279
Elastio strain recovery after stress release following steady flow ( r 3 Course of stress relaxation aftersuddenstrain[S(t) 1
(3/+) y
a:J
3 : J S_mm
1:-
*e-&
r * 0 d In
ACKNOWLEDGMENT
This work was supported in part by it grant from Research Corp. and in part by the research committee of the Graduate School of the University of Wisconsin from funds supplied by the Wisconsin Alumni Research Foundation. Grateful acknowledgment is also made of the fellowship in physical chemistry of the Carbide and Carbon Chemicals Division, Union Carbide and Carbon Corp. Since March 1951 these investigations have been part of a program of research on the physical structure and properties of cellulose derivatives and other polymers supported by the Allegany Ballistics Laboratory, Cumberland, Md., an establishment owned by the United States Navy and operated by the Hercules Powder Co. under contract NOrd 10431.
...
T
rWd In r
d In
Course of stress relaxation after cessation of p &pe-tjr d In steady flowa [3(t)] Subject to non-Newtonian correction ( 1 4 ) .
Y
T
xht JI,
-j
7
*
7
. .
(7)
the applications are carefully restricted to conditions where these expressions are valid. Over wide ranges of T , varies in a complicated manner, and it is believed preferable to rely on graphical methods using experimental data directly. Information concerning the form of the distribution function over wide ranges of time scale and its dependence on chemical composition, molecular weight, and molecular weight distribution should be forthcoming in the near future.
4.d ln
7
In
7
&d
’
NOMENCLATURE
Exact expressions for G‘, G”, and q ’ are derived from this by Equations 4a to 4c, and then the cycle is completed by substituting these expressions into the approximate Equations 5a-5c. The resulting values of a differ from the initial power law only by numerical factors which are functions of m. It may be assumed that the values obtained by applying Equations 5a-5c to experimental data are in error by the same factors, which can then be used to obtain the corrected second approximations ip
(-In
w) =
AG’(d log G‘/d log w )
(68)
@
(-In
w) =
BG”(1 - d log G”/d log w )
(6b)
(-In
W)
= -Cwq’(d
log
q’/d
log W )
(6c)
The correction factors are obtained as gamma functions of m, the negative slope measured on the log-log plot of the first approximation a against T (7)
B = C = (1
+ m)/2r (E - y) r (i + );
(7b)
It turns out that, for nt = 0.5, A = C = 0.90, which explains the fortuitous agreement shown in Figure 4 where m = 0.6. Equations 6a-60 should be good approximations except in regions where the value of m changes sharply. On the low T side of such a bend in the log-log plot of @ against T ) the value from G” or q’ (Equation 6b or 6c) will be most nearly correct; on the high T side of such a bend, the value from G’ (Equation 6a) will be preferable. From these considerations, it should be possible to obtain the function from dynamic data with adequate accuracy. A more detailed discussion, with tables of values of A and B, has been submitted for publication elsewhere ( 7 ) . APPLICATIONS OF €UNCTION m
When the function @ has been determined, i t may be used t o calculate various properties, including G’, G”, and v‘ as functions of frequency; the steady flow viscosity (for unvulcanized rubbers); the elastic energy stored in steady flow, as a function of shear stress; the extent of elastic recovery on release of stress following cessation of steady state flow; the course of stress relaxation following sudden strain; and the course of stress relaxation following cessation of steady flow. These are illustrated in Table I as both exact and approximate integrals. The latter and some of the former are easily evaluated graphically ( I C ) , though some of the exact forms would be tedious without computing aids. Attempts ( 4 ) to use simple analytical expressions for @ in formulas such as these may lead to quite erroneous results unless 906
a~ = reduction factor for relaxation times at temperature, 2’ A = second approximation correction factor for calculation of @fromG’ B = second approximation correction factor for calculation of %fromG” C = second approximation correction factor for calculation of @from7’ G’ = real part of the complex dynamic shear modulus G“ = imaginary part of the complex dynamic shear modulus Gi = contribution of ith mechanism to the instantaneous rigidity G: = G’ reduced to temperature TO Gi = G” reduced to temperature To k = constant in expression for % as a power function of T vz = negative slope of plot of log against log T t = time 2‘ = absolute temperature T O = reference temperature (usually 298’ K.) 3 = shearstress y = shearstrain i. = rate of shear strain r = gamma function q = steady flow viscosity = real part of the complex dynamic viscosity q: qu = q’ reduced to temperature TO p = density 7 = relaxation time 7p = relaxation time associated with Gi w = circular frequency (27r times frequency) w P = w reduced t o temperature TO @ = distribution function of relaxation times in shear ( @ d In T is the differential contribution to rigidity associated with relaxation times whose natural logarithms lie between In 7 and In 7 d In T )
+
LITERATURE CITED
( 1 ) Alfrey, T., and Doty, P., J . AppEied Phys., 1 6 , 7 0 0 (1946). (2) Andrews, R. D., IND.ENG.CHEM.,44, 707 (1952). (3) Conant, F. S., Hall, G. L., and Lyons, W.J., J . A p p l i e d Phys., 2 1 , 4 9 9 (1950). (4)
Dunnell, B. A., and Tobolsky, A. V., J . Chern. Phys., 17, 1001
(5) (6)
Ferry, J. D., J.Am. Chem. SOC.,7 2 , 3 7 4 6 (1950). Ferry, J. D., Fitagerald, E. R., Johnson, M. F., and Grandine,
(1949).
L. D., Jr., J . A p p l i e d Phys., 2 2 , 7 1 7 (1961). (7) Ferry, J. D., and Williams, M.L., J. Colloid Sci. (submitted). (8) Fitsgerald, E. R., Ph.D. thesis, University of Wisconsin, 1951. (9) Ivey, D. G., Mrowca, B. A., and Guth, E., J. A p p l i e d P h y s . , 2 0 , 4 8 6 (1949). (10) Kuhn, W., 2. p h y s . Chern., B 4 2 , 1 ( 1 9 3 9 ) . (11) Marvin, R. S., IND. ENG.CHEM.,44, 696 (1952). (12) Marvin, R. S., Interim Report Cooperative Program on Dynamic Testing, National Bureau of Standards (1951). (13) Nolle, A. W., J. P o l y m e r Sci., 5, 1 (1950). (14) Schremp, F. W., Ferry, J. D., and Evans, W. W., J . A p p l i e d P h y s . , 2 2 , 7 1 1 (1951). (15) ter Haar, D., J . P o l y m e r Sci., 6 , 2 4 7 (1951). (16) tar Haar, D., P h y s i m , 1 6 , 7 3 8 , 8 3 9 (1950). RECEIVED for review October 16, 1951.
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCBPTED Febuary 4,1952.
Vol. 44, No. 4
