Chapter 14
The Physics of Glassy Polycarbonate: Superposability and Volume Recovery 1
Paul A. O'Connell, Carl R. Schultheisz, and Gregory B. McKenna
Structure and Mechanics Group, Polymers Division, National Institute of Standards and Technology, Gaithersburg, MD 20899
The mechanical behavior of polymeric materials reflects the underlying physics of the mobility of the polymer chains. We have investigated the response of polycarbonate under three different conditions that will alter the polymer mobility: changes in temperature, strain and aging. We have examined the degree to which we can superpose the effects of these conditions to create master curves that can extend predictions beyond the range of feasible laboratory time scales. It was found that superposition could be successfully applied in each case, but that timetemperature superposition led to a master curve that was significantly different from that found using time-strain superposition. It was also found that curves fit to the [relatively short time] individual relaxation measurements at different temperatures could not be fit to the resulting master curve. Experiments were also performed to investigate the polymer mobility in volume recovery below the glass transition. It was found that volume equilibration required much longer than equilibration of the mechanical response in aging experiments, suggesting that simple free volume models may be inappropriate. In the vicinity of its glass transition temperature (T ), the mechanical response of a polymeric material is a strong function of both the rate of deformation and the temperature, with the modulus or compliance changing by several orders of magnitude as the Tg is traversed. These changes reflect the mobility of the polymer molecules, and indicate that the time scale of the experimental measurement is similar to the characteristic times associated with the relaxation mechanisms available to the polymer. Assuming that the same relaxation mechanisms are always active and that the characteristic time of each mechanism is affected in the same way by a change in g
*
Corresponding author.
©1998 American Chemical Society
199
200 temperature leads to the concept of time-temperature [or frequency-temperature] superposition (i). The most important result of superposition is that it allows for the creation of a master curve that can predict behavior well beyond typical experimental time scales. The first topic in this paper is an investigation into the time-temperature superposition of torsional stress relaxation data. It was found that the data could be superimposed, and that the stress relaxation data at each temperature could be fit with a [slightly different] stretched exponential function. However, the resulting master curve could not be fit with a stretched exponential function. The idea that the relaxation process reflects the mobility of the polymer molecules can be investigated using other means to change that mobility, such as the addition of a solvent or by applying a stress or strain. In this case, an applied torsional strain has been used to accelerate the deformation processes. Again, the assumption that each relaxation mechanism is affected identically suggests the possibility of timestrain superposition, and allows for the construction of a master curve. While timestrain superposition is applicable, the master curve is significantly different from the master curve obtained using time-temperature superposition. Finally, since the polymer relaxation processes are very slow below T , one must recognize that the polymer is not generally in an equilibrium state, but is evolving slowly toward thermodynamic equilibrium. This evolution has been examined directly through measurements of the volume, and is also reflected in changes in the mechanical response [modulus or compliance] in a process labeled physical aging (2). The changes in polymer mobility during aging [at constant temperature] again suggests the idea of time-aging time superposition, which allows for some prediction of the behavior at long aging times through a shift rate. However, direct measurement of the equilibration of both the volume and the stress relaxation behavior indicates that these properties equilibrate at different aging times, which suggests that the mechanical response is not directly coupled to the polymer volume. The material employed in all investigations was General Electric Lexan LS-2, a commercial, UV stabilized, medium viscosity grade Bisphenol-A polycarbonate (5). T was measured as 141.3 °C using differential scanning calorimetry heating at 10 °C/min (4). For the torsion experiments, the polycarbonate was obtained in the form of extruded rods of 25 mm diameter, while for the volume recovery measurements, pellets of the polycarbonate intended for injection molding were used. The pellets were approximately 3 mm in diameter by 5 mm long. g
g
Time-Temperature, Time-Strain and Time-Aging Time Superposition Stress Relaxation Experiments in Torsion. Cylinders of the polycarbonate were machined to a length of 50 mm and diameter 12 mm and a gauge section further machined of 30 mm length and 4 mm diameter. In order to remove the effects of previous thermal and/or mechanical history, the samples were heated to 145 °C [approximately 4 °C above the measured T J for 1 hour prior to testing. Residual birefringence was not observed on looking through crossed polars. The torsion measurements were carried out on a Rheometrics RMS 7200 (3) load frame, modified in our laboratory with a computer controlled servo-motor. The sample and grips were housed within a heater chamber for temperature control, with a measured oven stability [based on the range of measurements] of better than ±
201 0.1 °C. The torque force relaxations were measured from nominal strains γ [based on the cylinder outer radius] from 0.0025 to 0.08: γ = ΚΦ/L = ΛΨ
(1)
where Φ = angle of twist, R the cylinder radius, L the length of the gauge section and Ψ the angle of twist per unit length. Time-Temperature Superposition. Time-temperature superposition is widely used in the description of polymer behavior at temperatures above the glass transition temperature T (i), where the viscoelastic response function changes with temperature by a change in the time scale by a horizontal shift and in the intensity by a vertical shift b . An often-used representation for the response function in stress relaxation experiments is the stretched exponential of Kohlrausch-Williams-Watts (KWW) (5,6): g
T
G(f) = G exp[-(*/x/]
(2)
0
where G(t) is the shear modulus response at time ί, τ is a characteristic time, β a shape parameter related to the breadth of the relaxation curve, and G is the zero time shear modulus. A change in temperature from T to Τ results in a change in the characteristic relaxation time leading to a temperature shift factor Oj = τ (Το) /τ (Τ). Vertical shifts are seen as a temperature dependent zero time shear modulus G ; the vertical shift factor isb = GçfJ^/G (T). The modulus G is calculated from the torque by assuming that the stress and strain are both linear functions of the cylinder radius, as in Equation (1). Time-temperature superposition was applied to data generated from tests carried out at a given strain at temperatures between 30 °C and 135 °C. The data presented here are for a strain of 2%. The master curve was constructed as follows. The highest temperature [say TJ data were taken as the reference curve and the parameters [G , τ and β] to fit to the KWW function determined. Keeping β constant, the KWW fit to the next lowest temperature [TJ was found, and hence the temperature shift factor determined for the temperature change T T : 0
0
0
0
0
0
T
0
0
r
2
logfa/Γ,-Τ,)] = logKCT^ToCr,)]
(3)
l o g ^ T . - T , ) ] = loglGoCT^GoCT,)]
(4)
The T data were then re-fitted to the KWW function, this time allowing the β term to vary. Keeping this new β constant, the next lowest temperature (T ) data were fitted to the KWW function and the temperature shift factor determined for the temperature change T -T . By repeating this procedure for successively lower temperatures, it was possible to build-up the overalltime-temperaturemaster curve from the individual shifts at neighboring temperatures. The individual data sets and resulting master curve constructed using the above procedure are shown in Figure 1, from which it is evident that the data do appear to superimpose to form a master curve. 2
3
2
3
202 KWW Analysis of Temperature Data. In the following, we performed the KWW analysis in two ways. We first took the master curve obtained above and asked if it could be described by a KWW function. We then looked at the results from fitting KWW functions to the stress relaxation response at each temperature and asked what the apparent change in KWW parameters would be as a function of temperature. The time-temperature master curve of Figure 1, determined using the 'semi manual' method outlined above, is replotted in Figure 2 and compared to a KWW fit to these data [solid line]. It is immediately clear that the KWW equation does not adequately describe the relaxation behavior over the whole range of data. Next the KWW equation was applied to the data at each temperature, with no restrictions on the parameters, and the resulting β parameters are shown in Table I. The important observation here is the variation in the β parameter with temperature, which, if taken literally and on the assumption that the master curve is described by a KWW function, implies that time-temperature superposition does not apply to these data. However, above we have shown that a very good master curve representation of the data can be obtained by using time-temperature superposition. We interpret the result of the present analysis to show that the KWW parameters are not strongly determined using the limited time window [which is typical of mechanical tests] available here. Further, it shows the danger of interpreting limited data in terms of the KWW parameters. We also note that the master relaxation curve obtained from the time-temperature superposition was not well described by a KWW function. The temperature shift factors, log (%), required for superposition are shown in Figure 3. At low temperatures the shift factor is not highly temperature dependent, while at temperatures close to Tg the dependence is clearly very strong. The vertical shift factors required for time-temperature superposition are shown in Table II, referenced to 135 °C. The trend is the same as that observed for the time shift factors, with a relatively rapid change at temperatures a little below T followed by a leveling off at temperaturesferremoved from T . Although the vertical shifts between adjacent temperature curves is small, over a broad temperature range the cumulative effect can be significant, in this case leading to a maximum vertical shift of approximately 25% over the temperature range 30 °C to 135 °C. Note though that the vertical shifts are still small in comparison to the orders of magnitude by which the timescale is shifted. The shifts imply that the material is becoming intrinsically stiffer at lower temperatures. This may be reasonably expected, since from thermal expansion effects alone the material will be denser at lower temperatures. g
g
Time-Strain Superposition. The principle oftime-strainsuperposition is essentially the same as that for time-temperature superposition, though now there is a strain induced shift [acceleration] in the time scale of the material response (7-9). Again, within the context of the KWW function one can write thetime-strainshift function as γ (
