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Time dependent effects of pressure on the shear modulus of polypropylene
Time dependent effects of pressure on the shear modulus of polypropylene R. A. Duckett and S. H. Joseph Department of Physics, University of Leeds, Leeds LS2 9JT, UK (Received 15 September 1975) The time dependent changes in the shear modulus of polypropylene have been followed subsequent to both step and ramp changes in pressure for pressures up to 400 MN/m 2. Although the changes in pressure (P-jumps) are accompanied by transient heating effects which complicate the interpretation of the results; large changes in modulus can be seen at times of up to three hours after the pressure change. At these times the effects of temperature changes would be expected to be negligible. The results are consistent with the expected changes of free volume occurring during the dilatational creep under pressure, and lend no support to the 'microstress' theory recently proposed.
INTRODUCTION It is now well established that hydrostatic pressure can have a considerable effect on the physical properties of polymers. This has stimulated interest in relating the effects of pressure to those of temperature and time or frequency. Considerably less effort has so far been invested in studying the transient response of the polymer to changes in pressure, although this is also of considerable significance. There have recently been suggestions regarding the possible forms of the response to changes in pressure. One proposal 1-3 is that a change in pressure, the same as a change in temperature, induces a volume change which is accompanied by the generation of micro'stresses around structural heterogeneities such as crystallites. These microstresses combined with the usual type of non-linear behaviour produce an anomalously low value of the modulus. As the microstresses decay with time the modulus rises steadily towards the linear value appropriate to the new conditions, regardless of whether the change in temperature or pressure is positive or negative. A completely different approach suggests that changes in pressure cause changes in the free volume in the polymer, as the dilational creep progresses. These free volume changes will also affect most physical properties of the material and furthermore the 'sense' of the changes would be expected to depend on whether the pressure is increased or decreased. There are thus distinct qualitative differences between these two approaches. In an attempt to discern which of these applies to polypropylene we have investigated the time dependence of the large changes in the shear modulus following positive and negative changes in hydrostatic pressure (P-jumps) by performing torsion tests at constant twist rate, at different elapsed times from the P-jump. It turns out that the results of these experiments are complicated by the concomitant changes in temperature induced by the changes in pressure, in both the pressure transmitting fluid and in the polymer. In order to investigate the contribution to this temperature rise by the polymer itself, temperature changes due to adiabatic compression of polypropylene were studied using a constrained uniaxial compression cell. An alternative approach to the investigation of the time dependent shear modulus, involving the use of hydrostatic pressure increasing at constant ramp rates, was also used. This results in a closer approximation to iso-
thermal conditions, and provides more accurate temperature control. Although little previous work has been carried out directly on time dependent modulus changes following changes in pressure or temperature ~-3, the changes in modulus on the free volume scheme are expected to relate to total volume changes under such conditions. Such time dependent volume measurements have been made following small changes in temperature and pressure 4 and following large changes in temperature s in amorphous polymers near their dilatational glass transition temperature (Tg). Volume creep in several polymers has also been investigated at high pressures 6'7. There are more data available on the static effects of hydrostatic pressure on both shear and bulk moduli; such data have been reviewed s and we refer especially here to torsion pendulum data for polypropylene 9. EXPERIMENTAL
Torsion tests Full details of the apparatus and specimens will be presented elsewhere 1° so here we give only a brief description. The apparatus is a modified version of that previously described H. Right solid cylindrical specimens of isotropic isotactic polypropylene (PP), density 909.5 + 0.1 kg/m 3, are subjected to a constant rate of twist whilst under hydrostatic pressure applied via Plexol 201 lubricant (diethyl dihexyl sebacate). The specimen design minimizes the magnitude of end effects, thus enabling the strain in the specimen to be deduced accurately from the angle of twist of the grips, which can be monitored outside the pressure vessel. Torque is also continuously monitored via an angular transducer mounted on the torque tube, and the torquetwist curve for the specimen for small angles of twist is recorded on an X - Y recorder. From this curve the stressstrain curve for small strains under constant strain rate conditions may be deduced 1°. The vessel temperature is governed by a water bath controlled to -+0.3°C surrounding the vessel. The first two testing programmes followed the pressure schedule shown in Figure 1. Each step in pressure corresponded to 30 MN/m 2 in the first programme and to 100 MN/m 2 in the second. The specimen was loaded into the vessel and allowed to come to equilibrium at the testing
POLYMER, 1976, Vol 17, April
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Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph ~E 120 Z
400
pressure. In this way the whole o f the pressure schedule is completed. The results, 2% secant modulus values against elapsed time, at 58°C, are plotted in Figures 2 and 3. In the second pair of testing programmes the effects of starting at atmospheric pressure and increasing directly to constant pressures of 100, 2 0 0 , 3 0 0 and 400 MN/m 2 were examined (see Figure 4) at 39°C and 58°C. After application of the hydrostatic pressure the above time sequence of tests was performed; graphs of the modulus vs. elapsed time are shown in Figures 5 and 6.
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4 6 Ti me ( h ) Figure 4 Second pressure schedule, showing the pressure increased rapidly from atmospheric to each of 100, 200, 300 and 400 MN/m 2 in turn. The time at atmospheric pressure between tests was approximately 3600 sec
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Time {sec) Figure 5 S h e a r m o d u l u s v s . t i m e u n d e r p r e s s u r e ( l o g a r i t h m i c scale) according to second pressure schedule (see Figure 41. Pressures: A, 100 MN/m2; B, 200 MN/m2; C, 300 MN/m2; D, 400 MN/m 2. Nominal temperature 58°C D _
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temperature for at least 12 h at atmospheric pressure. A torsion test was then performed at a rate equivalent to a strain rate of 1.1 × 10-3/sec, up to a maximum strain of 2.5 x 10 -2. The specimen was immediately unloaded to zero torque and allowed to recover for 300 sec. The pressure was then changed to the first in the schedule (30 MN/ m 2 for the first programme), at a rate o f 5 MN/m 2 sec and a sequence of the above torsion tests performed at convenient logarithmically spaced times, i.e. at about 30 sec, 300 sec, 1000 sec, 3000 sec after the achievement of the new pressure. Then, after a further recovery period of 300 sec from the final test in the sequence, the pressure was changed to the next on the schedule, and the same sequence of torsion tests performed again, under the new constant
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P O L Y M E R , 1976, Vol 17, April
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Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph
Using appropriate engineering data values* for PP we get:
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Figure 7 Temperature changes due to P-jumps. The temperature was measured at the surface of the specimen using copper resistance thermometry. Pressure changes involved are: A, --100 MN/m2; B, --30 MN/m2; C, +30 MN/m2; D, +100 MN/m 2
Temperature effects In the hydrostaticfluid system.
Findley, Reed and Stem 6 have pointed out the need to consider temperature changes caused by adiabatic compression of hydrostatic pressure fluid. The temperature changes in our system are shown in Figure 7 for changes in pressure of-+100 MN/m 2 and -+30 MN/m 2. The temperature was found to rise at the rate of 0.09°C (MN/m2) -1 at pressures of up to 100 MN/m 2, the rate dropping slightly to about 0.06°C (MN/m2) -1 for pressures above 200 MN/m 2. This resulted in temperature changes of up to 25°C for a 400 MN/m 2 change in pressure. At all pressures, the temperature rise decayed with a time constant of approximately 340 sec as the centre of the vessel came into equilibrium with the bath. The temperatures above were measured using the change in resistance in a copper coil of resistance approximately 20 ohms wound in a single layer on a PP former. Thus the temperature measured was at the PP-fluid surface. In order to check that a significant amount of the polymer also experienced these temperatures, the temperature measurement was repeated after coating the coil with latex to a thickness of 1.5 mm. Similar temperature-time profiles were found as before, but delayed in time by about 40 sec. The resistance thermometry was checked against mercury-in-glass thermometers at atmospheric pressure, and had absolute accuracy better than +0.5°C. The pressure coefficient of resistance of copper 12 is such that the error in temperature measurement due to pressure changes with this method is less than 0.005°C (MN/m2) -1. Thus with this method we are able to find the temperature changes that have occurred within the specimen after the P-jump experiments. Within the polymer. In the case of PP, as with polymers in a rubbery state, the bulk modulus and density are of the same magnitude to those of oils, and there is a considerable temperature change on the application of pressure. It is readily shown la that for a reversible adiabatic change:
To examine this for PP we constructed, to fit an Instron compression/tension test machine, a simple pressure cell in which the specimen is restrained from lateral expansion during axial compression (see Figure8). The highest pressure used was 150 MN/m 2. The specimens were 13 mm diameter × 15 mm long and had a 34 swg chromel-alumel thermocouple inserted into a 0.9 mm diameter axial hole f'dled with grease, so that the junction was centrally placed within the specimen. In this mode of deformation the effective modulus of the specimen M can be shown to be: 4
G
M = K + -
where K is the bulk modulus and G the shear modulus. To check the operation of the cell, load deflection curves were plotted at low strain rate (6.4 x 10-5/sec) to ensure isothermal conditions, and the experimental value of M found for pressures less than 45 MN/m 2 was in good agreement with estimates based on published values of K and G at atmospheric pressure 7'1°. Above this pressure, M was seen to rise, an effect we attribute to an increase in K and G at high pressures offset by the slight decrease in G at high shear strains. (For very low loads, corresponding to pressures less than 15 MN/m 2, there is a region of low modulus where the specimen is effectively under unconstrained uniaxial compression before play is taken up and lateral strain is eliminated). To investigate the adiabatic heating effect intrinsic to the polymer, the following sequence of tests was performed: a nearly static load was applied to the specimen, and load cell and thermocouple outputs continuously recorded. * a = 3.3 × 10 - 4 (°C)-I at 20°C and Cp = 1.8 kJ/kg
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] h e constrained unia×ial compression cell used in conjunction with an Instron machine for compression beneath the cro~head. A. mild steel cylinder; B and C, mild steel push-rods; D, polypropylene specimen; E, thermocouple; F, Instron compression load-cell anvil; G, anvil attached to underside of moveable crosshead
P O L Y M E R , 1976, V o l 17, A p r i l
331
Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph
If we write IAT+_I= I+-hTp+ ATvl, then it can be seen from Figure 9 that ATp/AP decreases at higher pressures. This is as expected from the changes in a and Cp as pressure increases and the/3 transition is approached.
Pressure ramp experiments
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I
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Loading was achieved through a nominal strain rate of 6 x 10-3/sec corresponding to a maximum loading time of 10 sec. When the temperature once again became static (after approximately 1000 sec) the load was rapidly removed (unloading time ~0.1 sec). In Figure 9 the peak temperature change is plotted against the pressure change for both unloading and loading. We see a region at low stress corresponding to unconstrained uniaxial compression. Then tbr pressures greater than 15 MN/m 2 the temperature changes are linearly dependent on pressure, and equal and opposite for compression and decompression, with a temperature coefficient of 0.04°C (MN/m2) -1. There is a discrepancy between the temperature coefficient found here and that calculated above using standard engineering data values for Cp and a. This may be merely due to differences between samples of polypropylene, but may also reflect the shorter times involved in this experiment compared with those usually involved in measurements of a and Cp. Using the measured value of (a T/aP)s = 0.04 ° C(MN/m2) - 1 the low stress region of Figure 9 has also been analysed and the measured temperature changes are consistent with the interpretation of unconstrained uniaxial deformation. At stresses greater than 50 MN/m 2 the temperature changes for compression, IAT+I, exceed those observed for decompression, IAT_I. Further tests indicate that the values approach each other at low strain rates, extending the pressure range for which IAT+I ~ IAT_I. To assess the possibility that the divergence between IAT+I and ItxT_l is due to temperature changes arising from viscous work done, which being positive in sign, will increase IAT+I but decrease IAT_ I, we have calculated the temperature change ATv expected for a standard linear solid. Assuming that K is elastic, that the shear modulus G has an unrelaxed value Gu = 2100 MN/m 2 and a relaxed value Gr = 450 MN/m 2, and that the load is changed in a time very much smaller than the creep relaxation time, we find that ATv = 3.5 x 10 .5 (z~°) 2 (°C), where AP is the pressure change in MN/m 2. Values of IAT_I + IATvl are plotted in Figure 9 and are seen to lie approximately midway between the two curves for IAT+I and I/XT_I. Although the excellent agreement obtained may be somewhat fortuitous in view of the simplicity of the model assumed, it does seem to indicate that the difference between IAT+I and I/XT_I is due to viscous heating.
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POLYMER, 1976, Vol 17, April
We have shown that the rapid changes in pressure involved in the P-jump experiment necessarily implies transient heating effects of up to 25°C for P-jumps of 400 MN/m 2. In the next section we discuss the additional complexities of interpretation arising from this temperature change. To observe changes in modulus under more nearly isothermal conditions we have performed tests in which the pressure is increased at a constant rate and the modulus is measured at convenient intervals of pressure. (We refer to these as a pressure ramp experiments). In these experiments an initial rise in temperature of the vessel is seen, but the constant heat input due to the ramp results in an equilibrium temperature being obtained within 500 sec, at which temperature losses from the fluid to the surrounding water bath are equal to the heat input. Thus by presetting the bath temperature slightly below the desired experimental temperature by an amount calculable from the thermal characteristics of the vessel, we can achieve a well defined temperature which is effectively constant throughout the pressure ramp. With this pressure ramp experiment we have attempted to detect the effects of varying ramp rate within the range 67 and 1200 MN/m 2 h. At the higher rates it is possible that the pressure can be applied in times less than the volumetric creep relaxation time, whereas at low rates the polymer might be able to creep sufficiently fast to remain in equilibrium with the pressure at all times. In Figure 10 the results of such ramp experiments are shown, at tem-
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E Z
O
E
zL0
I
o
I
260
460
Pressure ( M N / m 2)
Figure 10 Shear modulus vs. pressure, from pressure ramp experiments. A t temperature = 21°C; A, 67 M N / m 2 h; I , 400 M N / m 2 h; A, 1207 M N / m 2 h. A t temperature = 58°C; I , 67 M N / m 2 h; O, 400 M N / m 2 h
Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph
O
Time Figure 11 Modulus changes expected following step changesin pressure according to the microstress theory. A, (--. --) compression or decompression assuming no temperature change; B, compression, allowing for initial temperature rise; C, decompression allowing for initial fall in temperature
peratures of 58°C and 21°C. (Where necessary small corrections for errors in temperature setting have been made.) It can be seen that at both temperatures the modulus increases with increasing pressure, and that there is a qualitative similarity to the effects of increasing frequency. Also, to within experimental accuracy, the only detectable effect of ramp rate has been at 58°C and below 400 MN/m 2 pressure, although the rate effects might have been expected to be more apparent in the glass transition region, say at 30°C, 200 MN/m z. DISCUSSION
Modulus changes on pressurization The data in Figures 2, 4, 5, 6 and 10 show clearly the development in time of the changes in shear modulus of PP subsequent to changes in pressure. The curves indicate that the hydrostatic pressure is taking the polymer through its glass (/3) transition. The data show that the changes in modulus are quantitatively consistent with the idea that the modulus is closely related to the volume of the specimen during loading and unloading; i.e. the application of pressure reduces free volume, and hence lengthens the relaxation times in the polymer. For PI' steps the fraction of the modulus change that occurs in the time interval 30 to 3000 sec increases with pressure, as one would expect on the basis that the higher pressures cause slowing down of the bulk creep process and hence the modulus change. The P~ steps show marked asymmetry compared to the PI" steps in that the process appears slower for pressure release than for pressure increase. However, the temperature changes resulting from the Pjump complicates any interpretation of the results. For times less than 1000 sec from the pressure change some sort of temperature correction is required. Findley et aL s corrected their hydrostatic creep data for temperature effects using the instantaneous temperatures and the coefficient of volume expansion. The efficacy of such a correction in our case is doubtful, both because of the increased theoretical complexity involved in dealing with dynamic pressure and temperature changes and their effects on modulus in the transition region, and also because of their
findings of its inability to describe the results from partial release of pressure experiments. Hutchinson and McCrum I proposed an alternative mechanism for time dependent changes in modulus in crystalline polymers following jumps in temperature or pressure. They proposed that the relaxation of microstresses generated by P-jumps would lead to a time dependent hardening as shown in Figure 11, curve A. However if we allow for the appropriate temperature changes we would expect curves B and C for Pt and P~ respectively. Comparison with Figure 2 indicates no support for the microstress hypothesis. It should be pointed out that PP does not show microstress effects in T-jump experiments either, unlike the polyethylene used by Hutchinson and McCrum l'a. A more helpful way of interpreting the P-jump results is to consider the temperature rise as partially negating the compression effects for a period up to 1000 sec. We can therefore attempt to interpret the data on the basis of an isothermal experiment in which the pressure is applied fairly gradually, and look to see if there are any changes in the modulus at times long compared with the effective rise time of the pressure. The data in Figures 2 and 3 show that as pressure increases, there is an increase both in the rate of change of modulus with pressure, and in the fraction of modulus change that takes place in the observed time interval, independent of the sign of the pressure change. This is consistent with the onset of the/3 transition as pressure increases. In Figures 5 and 6 we see the large modulus changes involved in the transition region, and also definite evidence of time dependence of modulus not only at times greater than 1000 sec but even at a time of 3 h, thus showing the need for working at a consistent time under pressure when investigating the dependence of modulus on pressure. A more quantitative analysis of these data is difficult in view of the temperature changes, not only as they instantaneously affect the modulus but also because temperature history can affect the modulus at a later time 1. However, the results here qualitatively agree with the ideas of temperature dependent bulk-viscosity as applied by Kovacs s'14 to his data on volume changes following a temperature change. As a transition temperature is approached (in the case of Kovacs, Tg for PVAC) the bulk processes slow down, as do shear processes, until bulk relaxation times are of the order of experimental times. If we further assume that the free volume affects the relaxation time and hence the modulus as in the WLF approach, then changes in modulus following P-jumps will become larger and slower with increasing pressure, as found in the present investigation. However, when we look at the pressure ramp experiments in this light, we would expect to find that as pressure rises the bulk relaxation time should increase until at a pressure P1 it becomes greater than the time allowed by the ramp rate. At pressures greater than PI the volume and hence the modulus should depend on the ramp rate, modulus being higher for lower ramp rate due to the greater degree of bulk contraction allowed by the longer times involved. Thus a bulk dispersion region should be revealed by these modulus measurements. Despite the slight dispersion seen at 58°C (Figure 10) there is none in the measurements at 21°C for which one would expect to see a higher dispersion as the transition region is swept through. Thus we conclude that at the rates we are able to employ, the shear dispersion region occurs at a lower pressure than the bulk dispersion region. This agrees with available evidence on bulk and shear relaxation times in glass forming liquids ~s and poly(vinyl
POLYMER, 1976, Vol 17, April
333
Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph acetate) (PVAC) 16'17which show that shear relaxation times are greater than bulk relaxation times. That is, in terms of the pressure ramp experiment, the bulk processes are relaxed throughout the shear 13transition region. Adiabatic heating
The adiabatic heating effect is of much greater experimental importance in studying modulus changes following a pressure jump than it is in studying volume changes. The adiabatic heating has been correctly ignored in volume measurements4. This is justified for PP as follows: equal changes in volume are produced by ~1 MN/m 2 pressure change and I°C temperature change at 20°C. However, the adiabatic heating effect produces only 0.04°C temperature change for 1 MN/m 2 pressure change. If the change in volume due to change in temperature following a P-jump is AVe, and the total change in volume consequent on the jump is AV, then AVe ~- O.04AV. In volume change experiments 4 the volume seen to relax in the experimental time scale is greater than 0.4 x AV, i.e. an order of magnitude larger than A Vo. However, in the case of modulus measurements the situation is different: equal changes of modulus are produced by 10 MN/m 2 and I°C s, and the adiabatic heatin~ produces a temperature change of 0.4°C for 10 MN/m z pressure change. Thus the difference between shear modulus changes resulting from adiabatic or isothermal pressure changes is much more significant that the difference between adiabatic and isothermal bulk moduli. The experimental precision achieved here in studying the adiabatic heating phenomenon could be much improved by the use of a pure hydrostatic pressure apparatus 6'7. This would enable the hulk processes in transition regions to be investigated via the temperature changes produced by application of pressure. Such investigations would hope to reveal both thermodynamic effects of changes in specific heat and thermal expansion coefficient as included in equation (1) on passing through the transition, and also viscous heating effects associated with the bulk compression.
hydrostatic pressure. In qualitative terms the effects on the shear modulus of increasing pressure are similar to those of decreasing temperature and take the polymer through its glass transition. The increases in modulus, which can occur over periods up to three hours after the pressure change, are consistent with a relaxation of free volume during the period of hydrostatic creep. The data lend no support to the 'microstress' hypothesis recently proposed. Subsidiary experiments have revealed that the pressure changes are also accompanied by significant temperature changes; no reliable method of correcting the data for these temperature changes can be proposed. ACKNOWLEDGEMENTS S. H. J. was supported by a Science Research Council studentship throughout this work. We are grateful to Dr J. S. C. Parry of Bristol University for the design and construction of the basic torsion apparatus. REFERENCES 1 2 3
Hutchinson, J. M. and McCrum, N. G. Nature 1974,252, 295 ParryJones, E. and Tabor, D. J. Mater. ScL 1974, 9,289 Hutchinson, J. M., Mathews, J. F. and McCrum, N. G. (1973).
4
Prec. 2nd Int. Conf. on Yield deformation and Fracture of Polymers. Cambridge, 1973 Rehage,G. and Goldbach, G. Bet. Bunsenges. Phys. Chem
5 6 7 8 9 10 11 12 13 14 15
CONCLUSIONS Considerable changes in the shear modulus of isotropic polypropylene have been observed subsequent to changes in
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POLYMER, 1976, Vol 17, April
16 17
1966, 70, 1144 Kovacs,A. J. Adv. Polym. Sci. 1963, 3, 394 Findley, W. N., Reed, A. M. and Stern, P. Mod. Plast. 1968, 45,141 Mallon,P. J. and Benham, P. P. Plast. Polym. 1972, 40, 77 Jones Parry, E. and Tabor, D. J. Mater. Sci. 1973, 8, 1510 Jones Parry, E. and Tabor, D. Polymer 1973, 14,617 Duckett, R. A. and Joseph, S. H. to be published Rabinowitz,S., Ward, I. M. and Parry, J. S. C. J. Mater. Sci. 1970, 5, 29 'International Critical Tables', McGraw-Hill,New York, 1935, Vol 6,p 137 Zemansky, M. W. 'Heat and Thermodynamics', McGraw Hill, New York, 4th Edn. 1957, p 248 Kovacs,A.J. Trans. Soc. Rheol. 1961,5,285 Litovitz, T. A. and Davies,C. M. in 'Physical Acoustics' (Ed. Warren P. Mason) 1965, Vol 2, p 281 McKinney,J. E. and Belcher, H. V. J. Res. Nat. Bur. Stand. (A) 1963, 67, 43 Williams,M. L. and Ferry, J. D. J. Colloid Sci. 1954, 9, 479
INTRODUCTION It is now well established that hydrostatic pressure can have a considerable effect on the physical properties of polymers. This has stimulated interest in relating the effects of pressure to those of temperature and time or frequency. Considerably less effort has so far been invested in studying the transient response of the polymer to changes in pressure, although this is also of considerable significance. There have recently been suggestions regarding the possible forms of the response to changes in pressure. One proposal 1-3 is that a change in pressure, the same as a change in temperature, induces a volume change which is accompanied by the generation of micro'stresses around structural heterogeneities such as crystallites. These microstresses combined with the usual type of non-linear behaviour produce an anomalously low value of the modulus. As the microstresses decay with time the modulus rises steadily towards the linear value appropriate to the new conditions, regardless of whether the change in temperature or pressure is positive or negative. A completely different approach suggests that changes in pressure cause changes in the free volume in the polymer, as the dilational creep progresses. These free volume changes will also affect most physical properties of the material and furthermore the 'sense' of the changes would be expected to depend on whether the pressure is increased or decreased. There are thus distinct qualitative differences between these two approaches. In an attempt to discern which of these applies to polypropylene we have investigated the time dependence of the large changes in the shear modulus following positive and negative changes in hydrostatic pressure (P-jumps) by performing torsion tests at constant twist rate, at different elapsed times from the P-jump. It turns out that the results of these experiments are complicated by the concomitant changes in temperature induced by the changes in pressure, in both the pressure transmitting fluid and in the polymer. In order to investigate the contribution to this temperature rise by the polymer itself, temperature changes due to adiabatic compression of polypropylene were studied using a constrained uniaxial compression cell. An alternative approach to the investigation of the time dependent shear modulus, involving the use of hydrostatic pressure increasing at constant ramp rates, was also used. This results in a closer approximation to iso-
thermal conditions, and provides more accurate temperature control. Although little previous work has been carried out directly on time dependent modulus changes following changes in pressure or temperature ~-3, the changes in modulus on the free volume scheme are expected to relate to total volume changes under such conditions. Such time dependent volume measurements have been made following small changes in temperature and pressure 4 and following large changes in temperature s in amorphous polymers near their dilatational glass transition temperature (Tg). Volume creep in several polymers has also been investigated at high pressures 6'7. There are more data available on the static effects of hydrostatic pressure on both shear and bulk moduli; such data have been reviewed s and we refer especially here to torsion pendulum data for polypropylene 9. EXPERIMENTAL
Torsion tests Full details of the apparatus and specimens will be presented elsewhere 1° so here we give only a brief description. The apparatus is a modified version of that previously described H. Right solid cylindrical specimens of isotropic isotactic polypropylene (PP), density 909.5 + 0.1 kg/m 3, are subjected to a constant rate of twist whilst under hydrostatic pressure applied via Plexol 201 lubricant (diethyl dihexyl sebacate). The specimen design minimizes the magnitude of end effects, thus enabling the strain in the specimen to be deduced accurately from the angle of twist of the grips, which can be monitored outside the pressure vessel. Torque is also continuously monitored via an angular transducer mounted on the torque tube, and the torquetwist curve for the specimen for small angles of twist is recorded on an X - Y recorder. From this curve the stressstrain curve for small strains under constant strain rate conditions may be deduced 1°. The vessel temperature is governed by a water bath controlled to -+0.3°C surrounding the vessel. The first two testing programmes followed the pressure schedule shown in Figure 1. Each step in pressure corresponded to 30 MN/m 2 in the first programme and to 100 MN/m 2 in the second. The specimen was loaded into the vessel and allowed to come to equilibrium at the testing
POLYMER, 1976, Vol 17, April
329
Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph ~E 120 Z
400
pressure. In this way the whole o f the pressure schedule is completed. The results, 2% secant modulus values against elapsed time, at 58°C, are plotted in Figures 2 and 3. In the second pair of testing programmes the effects of starting at atmospheric pressure and increasing directly to constant pressures of 100, 2 0 0 , 3 0 0 and 400 MN/m 2 were examined (see Figure 4) at 39°C and 58°C. After application of the hydrostatic pressure the above time sequence of tests was performed; graphs of the modulus vs. elapsed time are shown in Figures 5 and 6.
z
== 6C
t
E
E 2
2OO ~
t_
E
E
C el.
Time ( h i
Figure I
Pressure schedule tor the first two testing programmes (I) and (11). In each case step length, approximately 3600 sec. Pressure 'step-height' (I) 30 MN/m2; (11) 100 MN/m 2
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...SD
-- 3OO
3
z
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3
~
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°t
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=, -
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B
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I
t
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I0 4 Time {sec) Figure 2 Shear modulus vs. time under pressure (logarithmic scale) according to testing programme (I), (see Figure 1). Pressures: A, 30 MN/m2; B, 60 MN/m2; C, 90 MN/m2; D, 120 MN/m 2. e, pressure increasing; O, pressure decreasing I0 2
m
4 6 Ti me ( h ) Figure 4 Second pressure schedule, showing the pressure increased rapidly from atmospheric to each of 100, 200, 300 and 400 MN/m 2 in turn. The time at atmospheric pressure between tests was approximately 3600 sec
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temperature for at least 12 h at atmospheric pressure. A torsion test was then performed at a rate equivalent to a strain rate of 1.1 × 10-3/sec, up to a maximum strain of 2.5 x 10 -2. The specimen was immediately unloaded to zero torque and allowed to recover for 300 sec. The pressure was then changed to the first in the schedule (30 MN/ m 2 for the first programme), at a rate o f 5 MN/m 2 sec and a sequence of the above torsion tests performed at convenient logarithmically spaced times, i.e. at about 30 sec, 300 sec, 1000 sec, 3000 sec after the achievement of the new pressure. Then, after a further recovery period of 300 sec from the final test in the sequence, the pressure was changed to the next on the schedule, and the same sequence of torsion tests performed again, under the new constant
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P O L Y M E R , 1976, Vol 17, April
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Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph
Using appropriate engineering data values* for PP we get:
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Figure 7 Temperature changes due to P-jumps. The temperature was measured at the surface of the specimen using copper resistance thermometry. Pressure changes involved are: A, --100 MN/m2; B, --30 MN/m2; C, +30 MN/m2; D, +100 MN/m 2
Temperature effects In the hydrostaticfluid system.
Findley, Reed and Stem 6 have pointed out the need to consider temperature changes caused by adiabatic compression of hydrostatic pressure fluid. The temperature changes in our system are shown in Figure 7 for changes in pressure of-+100 MN/m 2 and -+30 MN/m 2. The temperature was found to rise at the rate of 0.09°C (MN/m2) -1 at pressures of up to 100 MN/m 2, the rate dropping slightly to about 0.06°C (MN/m2) -1 for pressures above 200 MN/m 2. This resulted in temperature changes of up to 25°C for a 400 MN/m 2 change in pressure. At all pressures, the temperature rise decayed with a time constant of approximately 340 sec as the centre of the vessel came into equilibrium with the bath. The temperatures above were measured using the change in resistance in a copper coil of resistance approximately 20 ohms wound in a single layer on a PP former. Thus the temperature measured was at the PP-fluid surface. In order to check that a significant amount of the polymer also experienced these temperatures, the temperature measurement was repeated after coating the coil with latex to a thickness of 1.5 mm. Similar temperature-time profiles were found as before, but delayed in time by about 40 sec. The resistance thermometry was checked against mercury-in-glass thermometers at atmospheric pressure, and had absolute accuracy better than +0.5°C. The pressure coefficient of resistance of copper 12 is such that the error in temperature measurement due to pressure changes with this method is less than 0.005°C (MN/m2) -1. Thus with this method we are able to find the temperature changes that have occurred within the specimen after the P-jump experiments. Within the polymer. In the case of PP, as with polymers in a rubbery state, the bulk modulus and density are of the same magnitude to those of oils, and there is a considerable temperature change on the application of pressure. It is readily shown la that for a reversible adiabatic change:
To examine this for PP we constructed, to fit an Instron compression/tension test machine, a simple pressure cell in which the specimen is restrained from lateral expansion during axial compression (see Figure8). The highest pressure used was 150 MN/m 2. The specimens were 13 mm diameter × 15 mm long and had a 34 swg chromel-alumel thermocouple inserted into a 0.9 mm diameter axial hole f'dled with grease, so that the junction was centrally placed within the specimen. In this mode of deformation the effective modulus of the specimen M can be shown to be: 4
G
M = K + -
where K is the bulk modulus and G the shear modulus. To check the operation of the cell, load deflection curves were plotted at low strain rate (6.4 x 10-5/sec) to ensure isothermal conditions, and the experimental value of M found for pressures less than 45 MN/m 2 was in good agreement with estimates based on published values of K and G at atmospheric pressure 7'1°. Above this pressure, M was seen to rise, an effect we attribute to an increase in K and G at high pressures offset by the slight decrease in G at high shear strains. (For very low loads, corresponding to pressures less than 15 MN/m 2, there is a region of low modulus where the specimen is effectively under unconstrained uniaxial compression before play is taken up and lateral strain is eliminated). To investigate the adiabatic heating effect intrinsic to the polymer, the following sequence of tests was performed: a nearly static load was applied to the specimen, and load cell and thermocouple outputs continuously recorded. * a = 3.3 × 10 - 4 (°C)-I at 20°C and Cp = 1.8 kJ/kg
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] h e constrained unia×ial compression cell used in conjunction with an Instron machine for compression beneath the cro~head. A. mild steel cylinder; B and C, mild steel push-rods; D, polypropylene specimen; E, thermocouple; F, Instron compression load-cell anvil; G, anvil attached to underside of moveable crosshead
P O L Y M E R , 1976, V o l 17, A p r i l
331
Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph
If we write IAT+_I= I+-hTp+ ATvl, then it can be seen from Figure 9 that ATp/AP decreases at higher pressures. This is as expected from the changes in a and Cp as pressure increases and the/3 transition is approached.
Pressure ramp experiments
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Loading was achieved through a nominal strain rate of 6 x 10-3/sec corresponding to a maximum loading time of 10 sec. When the temperature once again became static (after approximately 1000 sec) the load was rapidly removed (unloading time ~0.1 sec). In Figure 9 the peak temperature change is plotted against the pressure change for both unloading and loading. We see a region at low stress corresponding to unconstrained uniaxial compression. Then tbr pressures greater than 15 MN/m 2 the temperature changes are linearly dependent on pressure, and equal and opposite for compression and decompression, with a temperature coefficient of 0.04°C (MN/m2) -1. There is a discrepancy between the temperature coefficient found here and that calculated above using standard engineering data values for Cp and a. This may be merely due to differences between samples of polypropylene, but may also reflect the shorter times involved in this experiment compared with those usually involved in measurements of a and Cp. Using the measured value of (a T/aP)s = 0.04 ° C(MN/m2) - 1 the low stress region of Figure 9 has also been analysed and the measured temperature changes are consistent with the interpretation of unconstrained uniaxial deformation. At stresses greater than 50 MN/m 2 the temperature changes for compression, IAT+I, exceed those observed for decompression, IAT_I. Further tests indicate that the values approach each other at low strain rates, extending the pressure range for which IAT+I ~ IAT_I. To assess the possibility that the divergence between IAT+I and ItxT_l is due to temperature changes arising from viscous work done, which being positive in sign, will increase IAT+I but decrease IAT_ I, we have calculated the temperature change ATv expected for a standard linear solid. Assuming that K is elastic, that the shear modulus G has an unrelaxed value Gu = 2100 MN/m 2 and a relaxed value Gr = 450 MN/m 2, and that the load is changed in a time very much smaller than the creep relaxation time, we find that ATv = 3.5 x 10 .5 (z~°) 2 (°C), where AP is the pressure change in MN/m 2. Values of IAT_I + IATvl are plotted in Figure 9 and are seen to lie approximately midway between the two curves for IAT+I and I/XT_I. Although the excellent agreement obtained may be somewhat fortuitous in view of the simplicity of the model assumed, it does seem to indicate that the difference between IAT+I and I/XT_I is due to viscous heating.
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POLYMER, 1976, Vol 17, April
We have shown that the rapid changes in pressure involved in the P-jump experiment necessarily implies transient heating effects of up to 25°C for P-jumps of 400 MN/m 2. In the next section we discuss the additional complexities of interpretation arising from this temperature change. To observe changes in modulus under more nearly isothermal conditions we have performed tests in which the pressure is increased at a constant rate and the modulus is measured at convenient intervals of pressure. (We refer to these as a pressure ramp experiments). In these experiments an initial rise in temperature of the vessel is seen, but the constant heat input due to the ramp results in an equilibrium temperature being obtained within 500 sec, at which temperature losses from the fluid to the surrounding water bath are equal to the heat input. Thus by presetting the bath temperature slightly below the desired experimental temperature by an amount calculable from the thermal characteristics of the vessel, we can achieve a well defined temperature which is effectively constant throughout the pressure ramp. With this pressure ramp experiment we have attempted to detect the effects of varying ramp rate within the range 67 and 1200 MN/m 2 h. At the higher rates it is possible that the pressure can be applied in times less than the volumetric creep relaxation time, whereas at low rates the polymer might be able to creep sufficiently fast to remain in equilibrium with the pressure at all times. In Figure 10 the results of such ramp experiments are shown, at tem-
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Figure 10 Shear modulus vs. pressure, from pressure ramp experiments. A t temperature = 21°C; A, 67 M N / m 2 h; I , 400 M N / m 2 h; A, 1207 M N / m 2 h. A t temperature = 58°C; I , 67 M N / m 2 h; O, 400 M N / m 2 h
Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph
O
Time Figure 11 Modulus changes expected following step changesin pressure according to the microstress theory. A, (--. --) compression or decompression assuming no temperature change; B, compression, allowing for initial temperature rise; C, decompression allowing for initial fall in temperature
peratures of 58°C and 21°C. (Where necessary small corrections for errors in temperature setting have been made.) It can be seen that at both temperatures the modulus increases with increasing pressure, and that there is a qualitative similarity to the effects of increasing frequency. Also, to within experimental accuracy, the only detectable effect of ramp rate has been at 58°C and below 400 MN/m 2 pressure, although the rate effects might have been expected to be more apparent in the glass transition region, say at 30°C, 200 MN/m z. DISCUSSION
Modulus changes on pressurization The data in Figures 2, 4, 5, 6 and 10 show clearly the development in time of the changes in shear modulus of PP subsequent to changes in pressure. The curves indicate that the hydrostatic pressure is taking the polymer through its glass (/3) transition. The data show that the changes in modulus are quantitatively consistent with the idea that the modulus is closely related to the volume of the specimen during loading and unloading; i.e. the application of pressure reduces free volume, and hence lengthens the relaxation times in the polymer. For PI' steps the fraction of the modulus change that occurs in the time interval 30 to 3000 sec increases with pressure, as one would expect on the basis that the higher pressures cause slowing down of the bulk creep process and hence the modulus change. The P~ steps show marked asymmetry compared to the PI" steps in that the process appears slower for pressure release than for pressure increase. However, the temperature changes resulting from the Pjump complicates any interpretation of the results. For times less than 1000 sec from the pressure change some sort of temperature correction is required. Findley et aL s corrected their hydrostatic creep data for temperature effects using the instantaneous temperatures and the coefficient of volume expansion. The efficacy of such a correction in our case is doubtful, both because of the increased theoretical complexity involved in dealing with dynamic pressure and temperature changes and their effects on modulus in the transition region, and also because of their
findings of its inability to describe the results from partial release of pressure experiments. Hutchinson and McCrum I proposed an alternative mechanism for time dependent changes in modulus in crystalline polymers following jumps in temperature or pressure. They proposed that the relaxation of microstresses generated by P-jumps would lead to a time dependent hardening as shown in Figure 11, curve A. However if we allow for the appropriate temperature changes we would expect curves B and C for Pt and P~ respectively. Comparison with Figure 2 indicates no support for the microstress hypothesis. It should be pointed out that PP does not show microstress effects in T-jump experiments either, unlike the polyethylene used by Hutchinson and McCrum l'a. A more helpful way of interpreting the P-jump results is to consider the temperature rise as partially negating the compression effects for a period up to 1000 sec. We can therefore attempt to interpret the data on the basis of an isothermal experiment in which the pressure is applied fairly gradually, and look to see if there are any changes in the modulus at times long compared with the effective rise time of the pressure. The data in Figures 2 and 3 show that as pressure increases, there is an increase both in the rate of change of modulus with pressure, and in the fraction of modulus change that takes place in the observed time interval, independent of the sign of the pressure change. This is consistent with the onset of the/3 transition as pressure increases. In Figures 5 and 6 we see the large modulus changes involved in the transition region, and also definite evidence of time dependence of modulus not only at times greater than 1000 sec but even at a time of 3 h, thus showing the need for working at a consistent time under pressure when investigating the dependence of modulus on pressure. A more quantitative analysis of these data is difficult in view of the temperature changes, not only as they instantaneously affect the modulus but also because temperature history can affect the modulus at a later time 1. However, the results here qualitatively agree with the ideas of temperature dependent bulk-viscosity as applied by Kovacs s'14 to his data on volume changes following a temperature change. As a transition temperature is approached (in the case of Kovacs, Tg for PVAC) the bulk processes slow down, as do shear processes, until bulk relaxation times are of the order of experimental times. If we further assume that the free volume affects the relaxation time and hence the modulus as in the WLF approach, then changes in modulus following P-jumps will become larger and slower with increasing pressure, as found in the present investigation. However, when we look at the pressure ramp experiments in this light, we would expect to find that as pressure rises the bulk relaxation time should increase until at a pressure P1 it becomes greater than the time allowed by the ramp rate. At pressures greater than PI the volume and hence the modulus should depend on the ramp rate, modulus being higher for lower ramp rate due to the greater degree of bulk contraction allowed by the longer times involved. Thus a bulk dispersion region should be revealed by these modulus measurements. Despite the slight dispersion seen at 58°C (Figure 10) there is none in the measurements at 21°C for which one would expect to see a higher dispersion as the transition region is swept through. Thus we conclude that at the rates we are able to employ, the shear dispersion region occurs at a lower pressure than the bulk dispersion region. This agrees with available evidence on bulk and shear relaxation times in glass forming liquids ~s and poly(vinyl
POLYMER, 1976, Vol 17, April
333
Time dependent effects of pressure on polypropylene: R. A. Duckett and S. H. Joseph acetate) (PVAC) 16'17which show that shear relaxation times are greater than bulk relaxation times. That is, in terms of the pressure ramp experiment, the bulk processes are relaxed throughout the shear 13transition region. Adiabatic heating
The adiabatic heating effect is of much greater experimental importance in studying modulus changes following a pressure jump than it is in studying volume changes. The adiabatic heating has been correctly ignored in volume measurements4. This is justified for PP as follows: equal changes in volume are produced by ~1 MN/m 2 pressure change and I°C temperature change at 20°C. However, the adiabatic heating effect produces only 0.04°C temperature change for 1 MN/m 2 pressure change. If the change in volume due to change in temperature following a P-jump is AVe, and the total change in volume consequent on the jump is AV, then AVe ~- O.04AV. In volume change experiments 4 the volume seen to relax in the experimental time scale is greater than 0.4 x AV, i.e. an order of magnitude larger than A Vo. However, in the case of modulus measurements the situation is different: equal changes of modulus are produced by 10 MN/m 2 and I°C s, and the adiabatic heatin~ produces a temperature change of 0.4°C for 10 MN/m z pressure change. Thus the difference between shear modulus changes resulting from adiabatic or isothermal pressure changes is much more significant that the difference between adiabatic and isothermal bulk moduli. The experimental precision achieved here in studying the adiabatic heating phenomenon could be much improved by the use of a pure hydrostatic pressure apparatus 6'7. This would enable the hulk processes in transition regions to be investigated via the temperature changes produced by application of pressure. Such investigations would hope to reveal both thermodynamic effects of changes in specific heat and thermal expansion coefficient as included in equation (1) on passing through the transition, and also viscous heating effects associated with the bulk compression.
hydrostatic pressure. In qualitative terms the effects on the shear modulus of increasing pressure are similar to those of decreasing temperature and take the polymer through its glass transition. The increases in modulus, which can occur over periods up to three hours after the pressure change, are consistent with a relaxation of free volume during the period of hydrostatic creep. The data lend no support to the 'microstress' hypothesis recently proposed. Subsidiary experiments have revealed that the pressure changes are also accompanied by significant temperature changes; no reliable method of correcting the data for these temperature changes can be proposed. ACKNOWLEDGEMENTS S. H. J. was supported by a Science Research Council studentship throughout this work. We are grateful to Dr J. S. C. Parry of Bristol University for the design and construction of the basic torsion apparatus. REFERENCES 1 2 3
Hutchinson, J. M. and McCrum, N. G. Nature 1974,252, 295 ParryJones, E. and Tabor, D. J. Mater. ScL 1974, 9,289 Hutchinson, J. M., Mathews, J. F. and McCrum, N. G. (1973).
4
Prec. 2nd Int. Conf. on Yield deformation and Fracture of Polymers. Cambridge, 1973 Rehage,G. and Goldbach, G. Bet. Bunsenges. Phys. Chem
5 6 7 8 9 10 11 12 13 14 15
CONCLUSIONS Considerable changes in the shear modulus of isotropic polypropylene have been observed subsequent to changes in
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POLYMER, 1976, Vol 17, April
16 17
1966, 70, 1144 Kovacs,A. J. Adv. Polym. Sci. 1963, 3, 394 Findley, W. N., Reed, A. M. and Stern, P. Mod. Plast. 1968, 45,141 Mallon,P. J. and Benham, P. P. Plast. Polym. 1972, 40, 77 Jones Parry, E. and Tabor, D. J. Mater. Sci. 1973, 8, 1510 Jones Parry, E. and Tabor, D. Polymer 1973, 14,617 Duckett, R. A. and Joseph, S. H. to be published Rabinowitz,S., Ward, I. M. and Parry, J. S. C. J. Mater. Sci. 1970, 5, 29 'International Critical Tables', McGraw-Hill,New York, 1935, Vol 6,p 137 Zemansky, M. W. 'Heat and Thermodynamics', McGraw Hill, New York, 4th Edn. 1957, p 248 Kovacs,A.J. Trans. Soc. Rheol. 1961,5,285 Litovitz, T. A. and Davies,C. M. in 'Physical Acoustics' (Ed. Warren P. Mason) 1965, Vol 2, p 281 McKinney,J. E. and Belcher, H. V. J. Res. Nat. Bur. Stand. (A) 1963, 67, 43 Williams,M. L. and Ferry, J. D. J. Colloid Sci. 1954, 9, 479