Knowledge of the time- and temperature-dependent properties of polymers is essential for the design of structural components. Although numerous studies have addressed the time- and temperature-dependent shear response, there have only been a handful of systematic studies of the bulk deformation response.1–7 Early studies of the bulk relaxation response of polymers were performed in the 1950s and 1960s using various techniques, including forced compression, volume dilatometry, and ultrasonic studies.1–5 More recently, Deng and Knauss have performed measurements on poly(vinyl acetate) and poly(methyl methacrylate).6, 7 By comparing their bulk compliance data for poly(vinyl acetate) to the shear compliance data of several different researchers, they found that the transition from the glassy to the liquid response for the bulk compliance occurred primarily in the glassy regime of the shear response. This result indicates that the two responses arise from different molecular mechanisms as proposed by Leaderman,8 who suggested that the bulk modulus arises from intramolecular relaxations, whereas the shear modulus depends on intermolecular (or intersegmental) relaxations. The idea that the bulk and shear responses have different molecular origins is also backed up by the early data of Kono,5 as well as by calculations by Yee and Takemori,9 in which the bulk and shear dynamic moduli were calculated based on direct measurements of the dynamic tensile modulus and Poisson's ratio. Calculations of the bulk and shear moduli by Tschoegl10 also indicate that the bulk and shear (or tensile) relaxation functions generally differ in magnitude and shape.
Although much of the limited work in the literature is consistent with the dispersion in the bulk response occurring at shorter times or higher frequencies than the shear response, the issue is not completely resolved. Recent work by White and coworkers indicates that the time–temperature superposition shift factors for the bulk and axial responses are identical, based on measurements of the time-dependent Poisson's ratio during uniaxial elongation.11 Bero and Plazek12 also found that the bulk and shear retardation functions were identical (after a vertical shift) at short times although the shear response included longer time mechanisms not present in the bulk response. We note that Bero and Plazek measured the volumetric response to small temperature jumps in the glass transition region and from time–temperature superposition obtained a reduced curve that was related to the time-dependent bulk compliance; the need for small temperature jumps to ensure linearity of the volume recovery experiments was articulated. However, in early work by Kubat and coworkers,13, 14 they calculated the time-dependent bulk modulus from nonlinear physical aging data; more recently,15 they have called this parameter the aging modulus rather than the bulk modulus in recognition of the importance of linearity when one attempts to obtain the linear viscoelastic response.
In an effort to provide data to resolve the controversy surrounding the nature of the bulk and shear moduli, we perform pressure relaxation measurements for a polystyrene in two pressure ranges and compare these measurements to shear creep measurements made on the same material by Agarwal in the laboratory of Prof. D. J. Plazek.16 The paper is organized as follows. First, we briefly describe the material and methodology used in the measurements, and then we present results for the pressure dependence of Tg and the viscoelastic pressure relaxation responses measured as a function of temperature in two pressure ranges. The shift factors and retardation spectra, the latter of which are approximated from the relaxation spectra, obtained from the pressure relaxation master curves, are compared to those obtained by Agarwal and Plazek from the shear creep compliance. The master curves for the two viscoelastic functions are also compared. We end with a brief discussion of the results and conclusions.
The material studied is a polystyrene, Dylene 8, from Arco Polymers. It has a number-average molecular weight of 92,800 g/mol, and a weight-average molecular weight of 221,000 g/mol. The polystyrene sample was molded at 120 °C and annealed at 115 °C under vacuum. The annealed sample rods were machined down to a diameter of 4.0 mm and have a total mass of 1.242 g. The density of the sample after annealing in vacuum is 1.045 g/cm3 at 25 °C and 1 bar. This same Dylene 8 polystyrene has been used in a handful of studies in our laboratory, in the laboratory of Prof. D. J. Plazek, and in collaborative studies.16–20
A custom-built piston-type pressurizable dilatometer was used for measurements reported in this study. In this instrument, the polystyrene sample is surrounded by ˜5 g of fluorinated synthetic oil (Krytox® GPL107 from DuPont™) as confining fluid. According to the manufacturer, the oil's density is 1.95 g/cm3 at 0 °C and 1.78 g/cm3 at 100 °C. The instrument load cell (High Pressure Equipment Co.) is constructed of stainless steel 316 and has an inside diameter of 4.7 mm and a length of 102 mm, constituting a volume of ˜1.8 cm3. The system is pressurized by a piston made of hardened steel, whose position is controlled by a stepper motor with a resolution of 3.53 × 10−6 cm/step. The piston diameter is 3.16 mm; considering the backlash of the gear and the temperature stability, a conservative resolution of 7 × 10−6 cm3 can be achieved, which is much better than other PVT instruments.21–24 The piston position is measured by a linear variable differential transducer (LVDT, from Omega Engineering) mounted at the end of the piston. The instantaneous pressure of the system is monitored by a commercial pressure transducer (up to 60,000 psi, from Sensotec). The operational pressure range for this instrument is up to 250 MPa. A LabVIEW program has been written to automatically control the stepper motor (from Pacific Scientific) through the stepper motor controller (from Oriental Motor) and to record the temperature, pressure, and displacement of the LVDT. The temperature of the load cell which contains the sample is maintained by submerging the cell in a high precision oil bath (Hart Scientific), which is loaded with silicon oil and has isothermal temperature stability better than 0.01 °C at temperatures up to 250 °C. The temperature of the oil bath was calibrated against a NIST certified 1560 Black Stack (Hart Scientific) and is considered to be accurate to 0.01 °C.
Two types of measurements are reported in this work. Isobaric temperature scans were performed on cooling at 0.10 K/min from at least 30 °C above the pressure-dependent glass transition temperature, Tg(P), to at least 30 °C below it at various pressures. The glass transition temperature, Tg(P), is taken from these measurements to be the intersection of the extrapolated glass and liquid lines from LVDT voltage versus temperature curves. The slow cooling rate used for these measurements ensures that the temperature gradient in the sample is negligible during the runs.
In addition to temperature scans to obtain Tg(P), isothermal pressure relaxation measurements after volume jumps are performed at two starting pressures, Po = 30.2 ± 0.2 MPa and 60.5 ± 0.2 MPa, for various temperatures in the vicinity of Tg(Pmid), where Pmid is the midpoint pressure in the dispersion for the pressure relaxation master curves. Before each pressure relaxation experiment, the sample was heated and held at 15 °C above Tg(Po), where Po is the starting pressure before volume jumps, and then cooled down to the test temperature isobarically at 0.10 K/min. For runs above Tg, the temperature was held at the test temperature for at least 4 h before initiating the pressure relaxation experiment to ensure that the sample is at equilibrium density. For runs measured near and below Tg(Po), Struik's protocol25 is applied; that is, the test time is one-tenth of the waiting time (7–10 days) to ensure that physical aging does not affect the measurements. Pressure relaxation was measured as a function of time after volume jumps made in the linear viscoelastic response regime with volumetric strains of the polystyrene sample ranging from ˜0.5% up to 1.1%. In all of the jumps, the same apparent total volume change of 0.052 cm3 (including the confining fluid and the polystyrene sample and assuming no system compliance) was applied. Time zero for the measurements (when the relaxation starts, i.e., when t = 0 s) is taken to be the time when the pressure reaches the peak pressure immediately following the jumps. However, owing to a transition period resulting from adiabatic heating due to the jump and other system transient responses if any, pressure relaxation data are reported only for times after 102.5 s.
The perfluoropolyether oil (Krytox GPL107 from DuPont) surrounding the sample is chemically inert, has low volatility, and is stable between −30 and 288 °C. Swell tests on the polystyrene studied showed that no weight gain occurs upon immersion in the oil at 10 °C above its specific glass transition temperature for 15 days. Furthermore, isobaric cooling runs at 60.7 and 27.6 MPa were made before and after the pressure relaxation runs (8 months apart) and the Tg values from these runs agree within the error of the measurements. At 60.7 MPa, Tg prior to the pressure relaxation studies was measured to be 112.3 ± 1.7 °C (where the standard deviation is based on four cooling experiments before the jumps), whereas the Tg value measured after the pressure relaxation experiments was 112.6 °C; considering all five runs, the standard deviation of the measurements is taken to be 1.5 °C. At 27.6 MPa, Tg values of 102.5 °C and 101.4 °C were obtained before and after the pressure relaxation experiments, well within the standard deviation of the measurements.
We note that to eliminate the deleterious effect of adventitious gasses in the dilatometer, the oil was degassed at 230 °C in a vacuum oven at 0.05 bar for 24 h to eliminate dissolved gases and any other volatile components; in addition, before filling the system, it was evacuated by a vacuum pump at least 4 h to a vacuum of better than 6.7 Pa (0.05 Torr). Then the oil was pressurized into the system by a hand pressure generator (High Pressure Equipment Co.), and after filling, the system was re-evacuated for another 5 min.
The pressure relaxation measurements are analyzed as P(t) − Po in this work. This quantity is related to the viscoelastic bulk modulus K(t) of the polystyrene sample:
where Vsample is the volume of the sample and ΔVsample is the volumetric strain applied to the sample. The latter is related to the volume jump applied, ΔVapplied, and the volume and bulk modulus of the confining oil, Voil and Koil, respectively, assuming negligible compliance of the instrument:
Combining these equations, the relationship between the pressure relaxation and the bulk modulus of the sample is given by the following:
Since we measure pressure relaxation and the value of [P(t) − Po] changes during the experiment, the volume strain (ΔVsample) applied to the sample also changes in spite of the constant applied volume change; as already mentioned, the volumetric strain applied to the sample varies from ˜0.5% in the glassy state to 1.1% in the rubbery state. Hence, the quantity [P(t) − Po] is directly related to the apparent bulk modulus, and a Boltzmann superposition analysis must be applied to extract the true viscoelastic bulk modulus that would be obtained with the application of a constant volumetric strain.
The pressure-dependent glass transition temperatures, Tg(P), of polystyrene obtained from isobaric temperature scans performed on cooling at 0.10 K/min are shown in Figure 1. The glass transition temperature is taken as the intersection of the extrapolated glass and liquid lines in the LVDT versus temperature plot, an example of which is shown in the inset of Figure 1. As mentioned in the methodology section, the standard deviation in the Tg values is considered to be 1.5 °C based on the standard deviation of five isobaric cooling runs performed at 60.7 MPa. Our results are consistent with the Tg value measured at atmospheric pressure using capillary dilatometry for the same Dylene 8 polystyrene,17 shown as the unfilled circle in Figure 1. Tg is found to increase approximately linearly with increasing pressure in the pressure range studied (up to 220 MPa) with dTg/dP equal to 0.30 K/MPa. This value is in agreement with values reported in the literature for polystyrene which range from 0.25 to 0.40 K/MPa.26–32 Also shown in Figure 1 are comparisons of our data to those of Oels and Rehage33 and Quach and Simha.22 Oels and Rehage used a lower molecular weight polystyrene (Mw = 20,400 g/mol, Mw/Mn = 1.06); hence, their Tg values are ˜7 K lower than ours at low pressures. On the other hand, Quach and Simha used a similar polystyrene (Mw = 279,000 g/mol, Mn = 90,700 g/mol) and a similar cooling rate (0.167 K/min); the reason for the difference of ˜5 K at the lowest pressures between our results and theirs is not clear, although their results extrapolate to a Tg of ˜100 °C at atmospheric pressure, which is higher than might be expected for the cooling rate used.
The isothermal pressure relaxation curves as a function of time, P(t), are measured as a function of temperature in the vicinity of Tg(P) after volume jumps in two pressure ranges. For volume jumps made after a starting pressure of 60.5 ± 0.2 MPa, measurements are made from 89.0 to 131.0 °C, whereas for a starting pressure of 30.2 ± 0.2 MPa, measurements are made from 85.5 to 120.5 °C. The differential pressure relaxation curves, P(t) − Po, which are directly related to the apparent viscoelastic bulk modulus, are shown in Figures 2 and 3, respectively, for the two pressure ranges. In both figures, the responses at low temperatures indicate the bulk response of the glass, whereas at high temperatures, the bulk rubbery response is obtained. At intermediate temperatures, a time-dependent behavior associated with the viscoelastic bulk relaxation is observed. The good reproducibility of the data is indicated by the agreement of two runs at 113.5 °C shown in Figure 2, as well as by agreement of two runs at 118.75 °C shown in Figure 3. In both cases, the repeat runs are within the scatter of the data of approximately ±0.02 MPa.
Master curves for the data shown in Figures 2 and 3 can be obtained by time–temperature superposition of the data. However, a vertical shift is first needed to account for the temperature dependence of the bulk modulus. The pressure relaxation curves, plotted as aB(P − Po) versus logarithmic time after the vertical shift is applied, are shown in Figures 4 and 5, and the resulting master curves after application of the horizontal shift are shown offset to the right. Both master curves are plotted at a reference temperature of Tg(P), taking Tg to be that associated with the pressure at the midpoint of the dispersion based on the relationship shown in Figure 1.
The two pressure relaxation master curves are compared in Figure 6, again taking the reference temperature of each curve to be Tg(P). For ease of comparison, only a fraction of the data is shown. Although the magnitude of the dispersions differ for the two responses, with the change from the glass to the rubbery states being larger for the higher pressure range, the two dispersions occur on the same time scales and with the same shape. Thus, time–pressure superposition is valid for polystyrene over the pressure range studied. Also shown in Figure 6 are the fits of the master curves (lines) to the Kohlrausch–Williams–Watts34, 35 (KWW) function:
where Pg and Pr are the pressures corresponding to the glassy and rubbery responses, Po is the pressure before the jumps, τ is the relaxation time, and β is the Kohlrausch stretching exponent. The fitting parameters and their standard errors are shown in Table 1. It is notable that the best fits yield exponents of 0.29 and 0.28 for starting pressures of 60.5 and 30.2 MPa, respectively, very near the 1/3 scaling observed for Andrade creep36 in metals, amorphous polymers, and small molecule glass-formers.In metals, this scaling has been related to dislocation defects37, 38; on the other hand, in amorphous materials, Andrade creep is not well explained although it is the general observation.25, 39–42
Table 1. KWW Parameters for Master Curves
Po = 60.5 MPa
Po = 30.2 MPa
16.04 ± 0.001
12.51 ± 0.001
15.16 ± 0.001
11.81 ± 0.001
0.291 ± 0.001
0.277 ± 0.002
3585 ± 42
3316 ± 46
The logarithms of the vertical shift factors (aB) and horizontal shift factors (aT) are plotted against T − Tg in Figures 7 and 8, respectively. The vertical shift factors scale similarly with temperature (when plotted vs. T − Tg) independent of the pressure range in which the measurements were made. We find that aB is proportional to T0.4, reflecting the expected decrease in the bulk modulus with increasing temperature. On the other hand, for the shear modulus, one expects that the vertical shift factor in the rubbery state will scale as 1/T (neglecting density changes). We note, however, that no vertical shift factor was required to reduce the shear creep compliance data of this same Dylene 8 polystyrene.16
The horizontal shift factors from the pressure relaxation experiments (Fig. 8) similarly show the same temperature dependence independent of the pressure range of the experiments when plotted versus T − Tg. Furthermore, the horizontal shift factors from our measurements agree very well with those obtained for the shear data made by Agarwal16 (note that the Agarwal data were originally referenced to 100 °C and we shifted their data to Tg(P = 1 atm) = 93.8 °C in order that all three sets of data are compared at the Tg obtained at 0.10 K/min). Also shown is the fit of the Williams–Landell–Ferry43 (WLF) temperature dependence:
where the fitting parameters are found to be C1 = 16.2 and C2 = 30.4 by fitting Agarwal's data16 to Tg + 90 °C (not shown) taking Tg = 93.8 °C at 0.10 K/min. The data at the lowest measurements temperatures, below Tg −10 °C, level off due to the fact that the samples did not attain equilibrium density during the 7-day aging prior to the commencement of the pressure relaxation experiments. On the other hand, the change in slope to what appears to be a weaker temperature dependence at temperatures just below Tg (where equilibrium density is expected to be attained during the 7-day aging) is consistent with previous experimental44–49 and theoretical work,50–52 including our own experimental work on this same polystyrene.18, 20 We do note that in some reports47, 53, 54 no deviation from WLF behavior is observed at temperatures below Tg; a more detailed discussion of the equilibrium temperature dependence near Tg can be found elsewhere.47
To further compare the bulk and shear responses, we plot log[aB(P(t) − Po)] against the logarithm of the reduced time in the upper part of Figure 9 and plot the recoverable compliance of the shear response for the same polystyrene16 in double logarithmic scale below. The transition region in the bulk response occurs primarily in the short time region of the shear response and is finished by the time the Rouse modes become important in the shear response; this finding is consistent with the results of the dynamic modulus measurements on polystyrene and poly-(methyl methacrylate) by Kono5 as well as the dynamic compliance on poly(vinyl acetate) by Deng and Knauss6 although as discussed later, our interpretation of this finding differs from those of these researchers. In addition, when comparing the two responses, it is important to recognize that the change through the glass transition in bulk response is much weaker compared to that in shear response: the bulk modulus generally only differs by a factor of 2 or 3 through the glass transition region, whereas, the shear modulus changes 3 orders of magnitude through the glass transition region. Our comparison also qualitatively confirm the calculation predictions of Tschoegl.10
The relaxation time spectra, H(τ), are calculated from the pressure relaxation master curves using the second approximation method.55 To minimize the calculation error during differentiation, we use the KWW fit rather than the raw data to calculate the first and second derivatives required in the second-approximation method. The resulting relaxation spectra as a function of relaxation time in double logarithmic scale are shown in Figure 10 and agree well with one another, as expected based on the time–pressure superposition of the master curves. The retardation spectra, L(λ), can be calculated from the relaxation spectra base on the interconversion equation described by Smith.56 The retardation spectra are shown in Figure 11 along with that from shear from Agarwal.16 Our calculated retardation spectra and that from shear measurements agree reasonably for about four decades at short times but differ at long times, reflecting that long-time mechanisms are available to the shear response and not to the bulk response, consistent with the conclusion form Bero and Plazek.12
The observation that the bulk and shear measurements have similar time–temperature shift factors and similar retardation spectra at short times suggests that the molecular origin for bulk and shear responses are the same, in agreement with the conclusion of Bero and Plazek12 as well as White and coworkers'11 work. The fact that there are long-time chain mechanisms available to the shear response that are not available for the bulk response was also pointed out by Markovitz57 and by Berry and Plazek.58 Our results are not consistent with Leaderman's postulate8 that shear and bulk responses have different molecular origins, nor are they consistent with similar suggestions by Ferry,59 Kono,5 Knauss and coworkers,6, 7 Yee and Takemori.9
Another important finding from this work is that for this polystyrene in the pressure range studied, the primary effect of pressure is simply to shift the glass transition temperature to higher values, without a change in the width or temperature-dependence of the relaxation spectrum. Consequently, the horizontal and vertical shift factors show the same temperature dependence when plotted versus T − Tg(P). In addition, the placement and width of the relaxation spectrum obtained for the pressure relaxation curves is the same independent of the pressure range of the measurements when the reference temperature is taken to be Tg(P). This is also evident from the fact that the β parameters in the KWW fits of the two master curves are similar. On the other hand, the width of the dielectric relaxation has been found to increase,60 decrease,61 or change very little or not at all62–65 with increasing pressure; the discrepancies being perhaps due, in part, to overlap between the α- and β- (or β′-) relaxations, which might be expected to have different pressure dependences.64, 65 Other authors66, 67 have also suggested that the width of the relaxation region increases with increasing pressure based on dynamic light scattering studies. For mechanical properties, however, the effect of pressure on the α-relaxation is less well-studied. Moonan and Tschoegl68 have shown time–pressure superposition for shear relaxation, and McKinney and Belcher1 have shown the same for the bulk response. Our results are consistent with these previous works.
Isothermal time-dependent pressure relaxation measurements on a polystyrene were measured in two pressure ranges using a pressurizable dilatometer. The pressure-dependent glass transition temperature from isobaric cooling was also measured. Pressure relaxation master curves are constructed based on time–temperature superposition after vertical and horizontal shifts. The vertical shift factors are found to be independent of the pressure range of the measurements when plotted versus T − Tg(P). The horizontal shift factors are also found to be independent of pressure when plotted versus T − Tg, and they agree well with shift factors obtained from shear creep compliance measurements made on the same polystyrene at atmospheric pressure by Agarwal in the laboratory of Prof. D. J. Plazek. The similar shape of the pressure relaxation master curves, along with the fact that the temperature-dependent shift factors do not depend on pressure when plotted versus T − Tg, suggest that time–pressure superposition holds for the pressure range studied and that the primary effect of pressure is a shift in the glass transition temperature. Comparison of the pressure relaxation master curves and the shear creep compliance master curve (from Agarwal) confirms that the bulk relaxation occurs primarily in the short time region of the shear response. The retardation spectra from the bulk response agree with that from the shear response at short times but differ at long times. We conclude that the bulk and shear responses arise from similar molecular mechanisms at short times, but long-time chain mechanisms available to the shear response are not available to the bulk response. This conclusion is consistent with that of Bero and Plazek for an epoxy material.
We gratefully acknowledge funding from National Science Foundation grants, DMR 0308762 and 0606500. We thank G. B. McKenna and P. A. O'Connell at Texas Tech for their assistance in instrumentation. We also thank P. Bernazzani for initiating the design of the pressurizable dilatometer and D. J. Plazek for providing us with the data for the shear creep compliance response of Dylene 8.