DONALD J. PLAZEK
3480
Acknowledgment. We wish to thank Dr. J. G. Spencer for assisting with some of t’he measurements
and Mr. D. E. LaValle for preparation of the samples of KzReBr6.
Temperature Dependence of the Viscoelastic Behavior of Polystyrene
by Donald J. Plazek Mellon Institute, Pittsburgh, Pennsylvania
16615
(Received April 26, 1966)
The creep and recovery behavior of a narrow molecular weight distribution polystyrene sample was measured from 97.0 to 160’. Measurements were carried out on this 46,900 molecular weight sample in a creep apparatus that employs a magnetically levitated rotor in which constant torques are induced by means of a drag cup motor. Creep compliances measured extend from to above lo-* cm.2/dyne. Simple superposition of the results to obtain a reduced master curve was not possible because the viscous flow contributing to the total measured compliance had a different temperature dependence from that of the recoverable compliance. The recoverable compliance results by themselves were reducible. The temperature dependences obtained were analyzed in terms of free volume parameters.
Introduction I n 1941, Leaderman made the observation that the creep compliance curves obtained on a polymeric material a t different temperatures had the same shape and differed only in their positions on the logarithmic time scale.’* Tobolsky and Andrews in 1945 were the first to make use of this time-temperature superposition principle in constructing “master” curves over an enhanced time scale range with their stress relaxation data introducing at the same time the required rubberlike vertical or amplitude shift for amorphous polymers.lb The underlying assumptions for the successful application of reduced variables to linear viscoelastic phenomena were further developed by Ferry in 1950.lc Since then many reduced “master” curves for polymeric materials have appeared in the literature. I n some of these cases, it will be shown that the application of simple reduction principles was not appropriate although it appeared successful. I n a number of other studies2-6 it was found that simple reduction failed, but with the exception of the methacrylate polymers293 the viscoelastic response of amorphous polymers and their The Journal of Physical Chemistry
solutions have apparently proved to be reducible. The “master” curves have importance because the complete characterization of the time-dependent behavior requires description over the entire time scale. I n practice, the reduced curves come the closest to fulfilling this requirement. I n addition, the temperature dependence of the viscoelastic mechanisms is a by-product of the determination of such curves. For simple temperature reduction to be possible, all of the contributing viscoelastic mechanisms must have the same temperature dependence. I n the study reported here, the creep behavior of an anionically polymerized sample of polystyrene was (1) (a) H.Leaderman, Teztile Res. J., 11, 171 (1941); (b) A. V. Tobolsky and R. D. Andrews, J . Chem. Phys., 13, 3 (1945); (c) J. D.Ferry, J . Am. Chem. SOC.,72, 3746 (1950). (2) J. D.Ferry, W. C. Child, Jr., R. Zand, D. M. Stern, M. L. Williams, and R. F. Landel, J. Colloid Sci., 12, 53 (1957). (3) J. W. Beree. P. R. Saunder. and J. D. Ferry, ibid., 14, 135 (1959). (4)E.Catsiff, J. Offenbach, and A. V. Tobolsky, ibid., 11, 48 (1956). (5) H. Nakayasu, H. Markovitz, and D. J. Plaaek. Trans. SOC. Rheol., 5 , 261 (1961). I
TEMPERATURE DEPENDENCE OF
THE
VISCOELASTIC BEHAVIOR OF POLYSTYRENE
extensively measured at temperatures from 97", one degree below its conventional glass temperature, T,, to 160".
Experimental Section Materials. The polystyrene studied was prepareda by anionic means in vucuo with butyllithium as the catalyst.' The resulting whole polymer was subsequently fractionated6 by means of the conventional coacervation technique with methyl ethyl ketone as the solvent and methyl alcohol as the nonsolvent. From the fractionation results a ratio of weight-to-numberaverage molecular weight, ZW/Bn, was calculated to be 1.047. Three of the fractionation cuts with intrinsic viscosities that appeared to be within 0.5% of one another were mixed in solution and freeze dried from benzene. The three fractions represented that part from 2564% of the whole sample. The Flory temperature intrinsic viscosity, [qIe, for the combination was determined6 in cyclohexane a t 34.5" to be 0.184. I t is believed that BW/Rn for the combined central fractions is about 1.01. A molecular weight of 46,900 was calculated from [ V I @ using the expressiong [?IB = 8.5 X 10-4M0.5. The relation = 0.44[~]f::L~~ was used to obtain the corresponding intrinsic viscosity in benzene a t 30" so that a molecular weight of 45,300 could be obtained from the Ewart and Tingey relationlo for comparison with earlier characterization measurements in the literature. Reagent grade cyclohexane was dried overnight over cdcium sulfate, filtered, and distilled under Nz through a packed column designed to 80 theoretical plates a t total reflux. The central fraction which distilled a t a constant temperature and had been stored in a dark bottle under Nz was used in the intrinsic viscosity determination. Method. Torsional creep and creep recovery measurements were made with an instrument that contained a magnetically suspended rotor. The electronic feedback circuit employed to achieve stable suspension was developed by Dr. Victor MacCosham.ll Constant torques were induced in the rotor by means of a drag cup motor, and angles reflecting the torsional deformation in the cylindrical samples were monitored with a light lever and a Beckman photopen recorder modified to obtain a range of chart speeds from 30 to 960 in./hr. Temperatures were held constant to within 0.05" with a silicone oil bath (Dow Corning 550). Cylindrically shaped samples, 0.64 cm. in diameter and about 0.64 cm. high, were prepared in a vacuum mold at abouz 150". Samples that were measured were visible a t all times
3481
through a pair of mutually perpendicular flat glass windows in the sample housing of the instrument and corresponding windows in the stainless steel thermostat. Sample heights were measured optically to within a few microns with a Gaertner traveling microscope equipped with a relay lens system to yield 50 power a t a working distance of 19 cm. Sample coefficients, j / h , where j is the second moment of the crosssectional area and h is the height of the right circular cylinder, were calculated from the expression m2/ 2irp2h3, where m is the mass of the sample and p is I
I
I
-4
I
I
I
I
I
-
144.90c
I
-10 1
2
I
I 3
I
I 4
, l
i
5
Log t (see.)
Figure 1. The creep compliance Jp(t),cm.*/dyne, of polystyrene, M , = 46,900,plotted logarithmically against the time, sec. Temperatures of measurement are indicated. Subscript p indicates amplitude adjustment for the temperature dependence of the rubberlike nature of the response.
(6) This sample was prepared and pursed by Timothy Altares, Jr. Its fractionation was carried out by Marguerite Fulton. (7) (a) D. J. Worsfold and S. Bywater, Can. J. Chem., 38, 1894 (1960); (b) F. Wenger and S.-P. 8. Yen, Makromol. Chem., 43, 1 (1961). (8) This measurement was made by Elkabeth Frommell. (9) T. Altares, Jr., D. P. Wyman, and V. R. Allen, J . Polyner Sci., A2,4633 (1964). (IO) R. H. Ewart and H. C. Tingey, paper presented a t the 111th National Meetinlg of the American Chemical Society, Atlantic City, N. J., 1947. (11) V. J. MaoCosham, "Conference on the Ultracentrifuge," Academic Press Inc., New York, N. Y., 1963, p. 249.
Volume 69,Number 10 October 1966
DONALD J. PLAZEK
3482
To make meaningful investigations a t any temperature below a materid's T,, one must take pains to ascertain the volume of the sample at the time of measurement or at least record its thermal history. Viscosities can be determined in four different ways, (1) After apparent steady-state flow is reached, the viscosity, q, can be calculated from the velocity of deformation. (2) Following a reasonably long creep Results and Analysis run, the difference between the early parts of the creep The creep compliance results obtained are logarithand recovery portions of the run can be shown to be a mically plotted as a function of the logarithm of the measure of the viscous deformation through the applitime, t in seconds, in Figure 1as Jp(t)= (Tp/Topo)J(t), cation of Boltzmann superposition. (3) Determinacm.2 dyne, where p is the density at the temperature, tion can be made of the amount of permanent deformaT'K., of measurement and po is the density at the tion produced during a creep run by waiting for comreference temperature, To, 373.2"K. The vertical plete recovery. An increase in temperature to speed adjustment Tp/Topo takes into account the temperathe recovery is necessary if the creep run was longer ture dependence of rubberlike nature of the response. than a few minutes or the recovery time becomes exMeasurements were made from 97.0 to 160.0". At cessively long. (4)Ninomiya has shown14 that a plot 160" the response was experimentally entirely viscous of (J(t)/t)(d log J(t)/d log t) vs. l / t yields a relatively in nature and therefore is not shown in this compliance linear extrapolation to the intercept. This intercept is plot. Most of the curves are composites of two or the limiting value of dJ(t)/dt at infinite time, which is three runs. Besides two separate installations, in the reciprocal of the viscosity. situ manipulations of the sample shape along with the All four methods were used in obtaining the viscosities use of torques which varied from 75 to 3,450 dynes/cm. listed in Table I. According to the reduction prinwere necessary to measure accurately the millionfold ciples,lCif all of the contributing mechanisms have the change in compliance. At compliances less than same temperature dependence, all of the creep data should fall onto a single curve when log J,(t) is plotted 10-8 cm.Z/dyne the sample had to be drawn into a longer thin cylinder (diameter h.0.2 cm., height ~2 as a function of log [tq(TO)/qP(T)], where qp(T) = cm.). Sample drawing was carried out at 145" dT)TOPO/TP. (a* = ?P(T)/dTO) = j P ( t ) T O / j P ( t ) T where most of the deformation was viscous. The where Jp(& = J,(&; dot denotes time derivative.) sample was allowed to relax at the high temperature for about 0.5 hr. to eliminate orientation in the sample. Table I : Temperature Dependences" Upon removal of the drawn-out sample from the instrument at, room temperature, inspection in a polarizing microscope revealed no perceptible birefringence and hence negligible orientation in the central highly 0 (0.81) 97.0 (12.353) 1.13 0.994 elongated section. A trace of birefringence a t the 100.0 (11.540) 0 0 1.000 ends of the sample was probably caused by the cooling 100.6 (11.39) ( - 0.15) -0.22 1.001 to room temperature with the sample adhering to the 101.8 (11.10) (-0.44) -0.585 1.004 -1.04 10,503 104.5 stainless steel sample surfaces. Since the diameters of 1.010 -1.385 106.7 -1.50 1.014 -1.96 10.039 the drawn-out samples were not uniform, an em-2.07 109.5 -2.58 9.474 1.020 pirical sample coefficient had to be determined by 114.5 -3.49 8.680 -2.86 1.031 measuring and matching the level of compliance to a 125.0 -4.195 -4.87 7.345 1.052 previously determined curve. -5.105 133.8 -5.67 1.070 6.435 144.9 -5.98 (-6.41) 1.093 5.523 It can be seen in Figure 1 that the creep compliance -6.98 160.0 1.123 4.556 (-7.13) of this 46,900 molecular weight polystyrene has been measured from that of glassy hardness, cm.2/dyne, a Quantities in parentheses have been calculated using equations given in text. a t 97.0" to that of a very viscous liquid a t 145" and above. The conventional glass temperature, Tg, at this molecular weight is 97.7",13 so that at 97.0" we can be confident that our period of thermostating, (12) T.G Fox and 5.Loshaek, J . POZ~VWT Sci., 15,371 (1955). (13) T.G Fox and P. J. Flory, ibid., 14, 315 (1954). about 10 hr. before the start of the determination, (14) K. Ninomiya, J . Phys. Chem., 67, 1152 (1963). ensured thst the sample was at its equilibrium volume.
its density ( g . / ~ m . ~a)t the temperature of measurement. Creep compliances, J ( t ) , cm.2/dyneJ were computed from ja/h. The angle of twist, a, is measured, and the applied torque, 7, dyne-cm., is known from a previous calibration. Densities were calculated using 1/p = 0.767 5.5 X lO-*T 643 X T I M , Tis in "K.12
+
The Journal of Physical Chemistry
+
TEMPERATURE DEPENDENCE OF
THE
VISCOELASTIC BEHAVIOR OF POLYSTYRENE
3483
Log tvp(lOOoC)/~ptT) ) , = 100'. Figure 2. Logarithmic plot of creep compliance, J,(t) against reduced time scale t / a T , a T = q p ( T ) / q p ( T oTo Failure of temperature reduction is indicat'ed. Long-dashed line is the 100" t / q contribution. Short-dashed line is the reduced recoverable compliance curve.
Figure 2 illustrates clearly that a single curve is not obtained when such an attempt is made to reduce the data to 100'. The data have merged at the long time, high temperature end, where they must because here J ( t ) is largely determined by the viscous contribution, t/v. I n the region of the glasslike to rubberlike or primary transition, the separation of curves is pressing toward an order of magnitude. The 100' viscous deformation is represented by the long dashed line. According to current phenomenological theories of linear viscoelasticity, creep compliance for a linear amorphous polymer is described as
J(t)
= J,
+J 4 t ) +
t/rl
(1)
where Jg,the glassy compliance, is a constant in the neighborhood of cmS2/dyneand represents the instantaneous contribution to the deformation by the bending and stretching of molecular bonds; J,, a
constant, is the steady-state or long-time-limiting recoverable compliance; $(t) is a normalized retarded elasticity function, which is zero at t = 0, describing the form of the time-dependent recoverable deformntion. The logarithmic recoverable compliance curves, log (J,(t) - t/vp), shown in Figure 3 as functions of log t have been obtained directly from recovery measurements following fairly long creep runs or indirectly by subtracting the viscous contributions, t/v, from the measured creep compliance results. Reliable shift factors, uT, from 97.0 through 133.8" were obtained and are given in Table I. Utilizing these t,emperature shift factors, which were obtained only from the retarded elastic behavior, the curves from Figure 3 were reduced to 100". The resulting curve is shown in Figure 4. It is clear that superposition has been achieved and that the retarded elastic v o l u m e 69, N u m b e r 10 October 1966
DONALD J. PLAZEK
3484
4
- I
-
n F
1 I
- -E c
h
a
7
Y
0
0 -I
-c
-1c
Figure 3. The logarithm of the recoverable compliance, J,(t) indicate temperature of measurement; identzed in Table I.
- t/vp, shown aa a function of logarithmic time.
behavor has a different temperature dependence from the viscous flow. This “master” curve describes the behavior of the recoverable compliance at lOO”, extending from a level of glassy hardness, through the primary, glasslike to rubberlike transition to a smaller second transition in the usual “plateau” region. Whereas the primary transition in this bulk material reflects the retarded configurational adjustments of the individual polymer chain backbones to the applied tress,^^^^^ the plateau cm.2/dyne has existing in the neighborhood of been attributed to the existence of an entanglement network.” The final increase seen, starting at about 108 sec. on the reduced time scale, has been attributed to the slipping of the chain entanglements. However, on the basis of the response exhibited by poly(dimethy1)siloxanel8 and polyvinyl acetate1$ (both of which The Journal of Physical Chemistry
Pips
exhibit at least two transitions, beyond the rubberlike plateau), it is certain that even a complete qualitative picture for this region is lacking. It is likely that the apparent reduction of the second transition seen here is spurious and that more accurate measurements would reveal a temperature behavior different from that of the primary transition and of flow. In addition, we wish to point out for the theorist’s consideration that experimentally the compliances, (15) (a) P. E. Rouse, Jr., J . Chem. Phys., 21, 1272 (1953); (b) F. Bueohe, ibid.. 22, 603 (1954). (16) J. D.Ferry, R. F. Landel, and M. L. Williams, J. Appl. Phys., 26, 359 (1955).
(17) F. Bueohe,
