Time-temperature superposition plots

Figure 3.4 Creep plot for T0 obtained using time-temperature superposition. (After J. Fried, Plastics Engineering, July 1982, with permission.)...

Because of the uncertainties involved in the decomposition, this procedure would not appear to be a practical way to determine the AHa value needed for Equation 8. It does, however, demonstrate three important points (1) it is the compliances of the mechanisms that are additive (2) T0 and AHa can be obtained from plots such as those shown in Figures 7 and 8 of shift data determined in either relaxation or creep experiments without decomposition of compliance master curves (3) Equation 8 describes time-temperature superposition in Kraton 102 adequately within the experimental accuracy. 

The effects of strain rate and temperature are correlated, and can be modeled (Kinloch and Young, 1983, Kinloch, 1985). For different temperatures and strain rates, GIc and the time to failure, tf, were measured. Using the time-temperature superposition principle, shift factors (aT) applicable to the time to failure tf, were determine. Shift factors plotted against (T — Tg) are independent of the type of test used (Fig. 12.14). The construction of a typical master curve GIc versus tf/aT is shown in Fig. 12.15 (Hunston et al., 1984). The value of GIc may be predicted for any strain rate/temperature combination. This model can also be applied to rubber-modified epoxies (See chapter 13). 

For semi-crystalline polymers with melting points of more than 100 °C above the glass transition temperature and for amorphous polymers far above the glass transition temperature Tg (at around T = Tg + 190°C), the shift factors obtained from time-temperature superposition can be plotted in the form of an Arrhenius plot for thermally activated processes ... 

Figure 3.13 shows the shift factors aT determined from time-temperature superposition as a function of temperature for melts of two semi-crystalline thermoplastics as well as the Arrhenius plot. For the two polyethylenes (HDPE, LDPE), the progression of log ax can be described with the Arrhenius equation. The activation energies can be determined from the slope as Ea(LDPE) 60 kj/mol and Ea(HDPE) 28 kj/mol. Along with polyethylenes (HDPE, LDPE, LLDPE), other significant semi-crystalline polymers are polypropylene (PP), polytetrafluoroethylene (PTFE) and polyamide (PA). 

Figure 3.13 Left Shift factors aT from time-temperature superposition of two semi-crystalline thermoplastics [13]. Right Arrhenius plot a(T)=f(1/T). Lines Arrhenius Eq. 3.14 with Ea,HDPE=28 kj/mol and EaLDPE=60 kj/mol...

As an example of the concentration dependence of viscoelastic properties in Fig. 16.11 the shear creep compliance of poly(vinyl acetate) is plotted vs. time for solutions of poly(vinyl acetate) in diethyl phthalate with indicated volume fractions of polymer, reduced to 40 °C with the aid of the time temperature superposition principle (Oyanagi and Ferry, 1966). From this figure it becomes clear that the curves are parallel. We may conclude that the various may be shifted over the time axis to one curve, e.g. to the curve for pure polymer. In general it appears that viscoelastic properties measured at various concentrations may be reduced to one single curve at one concentration with the aid of a time-concentration superposition principle, which resembles the time-temperature superposition principle (see, e.g. Ferry, General references, 1980, Chap. 17). The Doolittle equation reads for this reduction ... 

Demonstration of the time-temperature superposition principle, using oscillatory shear data (G, filled circles and G", open diamonds) on a PVME melt with M — 124000 gmol. The right-hand plot shows the data that were acquired at the six temperatures indicated, with Tg = - 24°C chosen as the reference temperature. All data were shifted empirically on the modulus and frequency scales to superimpose, constructing master curves for G and G" in the left-hand plot. Data and... 

Mechanical relaxation experiments reveal that relaxation of shear moduli are similar to those of electrical moduli (chapter 7). In general mechanical relaxation spectra (G or E) exhibit a higher value of FWHM compared to M (electrical relaxation spectra). The mechanical relaxation data can also be collapsed on to master plots, which suggests that they obey time temperature superposition principle. There is therefore a parallel theoretical basis for the phenomenon. 

The principle of time-temperature superposition is that there is a temperature-shift factor that allows all data to be plotted on a master curve. This presupposes that there is no change in mechanism during the reaction, so that Equation (3.8) applies (Prime, 1997b). Such superposition processes are regularly used in rheology and the WLF equation is routinely applied when the system is above temperature Tg. When the system is controlled by... 

There is growing evidence that t-T superposition is not valid even in miscible blends well above the glass transition temperature. For example, Cavaille et al. [1987] reported lack of superposition for the classical miscible blends — PS/PVME. The deviation was particularly evident in the loss tangent vs. frequency plot. Lack of t-T superposition was also observed in PI/PB systems [Roovers and Toporowski, 1992]. By contrast, mixtures of entangled, nearly mono-dispersed blends of poly(ethylene-a/f-propylene) with head-to-head PP were evaluated at constant distance from the glass transition temperature of each system, homopolymer or blend [Gell et al, 1997]. The viscoelastic properties were best described by the double reptation model , viz. Eq 7.82. The data were found to obey the time-temperature superposition principle. 

The fourth step is to plot the shift factors Aj against temperature (Figure 2.15c). This representation of the time-temperature superposition characteristic of viscoelastic materials has been extensively analysed with the well-known Williams-Landel-Ferry (WLF) relationship, at temperatures above T ... 

For the moduli data, the time-temperature superposition fails at intermediate (0 between the segmental and global relaxation processes because these processes exhibit different Ojq at low T - (see, e.g., Adachi and Kotaka, 1993 Inoue et al., 1991, 1996 Kremer and Schonhals, 2003). (This failure is not well resolved in the compressed scale of the plots shown in Figure 3.3.) The superposition works separately at high and low ca where the viscoelastic data are dominated by one of these processes. In contrast, the dielectric data satisfy the superposition in the entire range of co because those data detect just the segmental relaxation process, although it fails in a close vicinity of... 

Data Analysis. PeakFit version 2.0 from AISN software, Jandel Scientific (Corte Madera, California) was used to separate overlapping transitions in tan(S) and e" versus temperature plots. TA Instruments model 2000 thermal analyzer was used with version 4.1 time-temperature superposition software to analyze the stress data. 

Our interest lies in the loci of failure points as a function of temperature. The family of curves of stress to break (t, at various temperatures (of which the dashed-line portion of Figure 1.22 is representative) is plotted schematically in Figure 1.23 (Scott, 1967). Use may now be made of the time-temperature superposition principle and the WLF equation (Section 1.5.7) to construct a master curve, as shown in Figure 1.24. 

The molecular theory predicts strong temperature dependenee of the relaxation ehar-acteristics of polymeric systems that is described by the time-temperature superposition (TTS) principle. This principle is based on numerous experimental data and states that with the change in temperature flie relaxation spectrum as a whole shifts in a self-similar manner along t axis. Therefore, dynamie functions corresponding to different temperatures are similar to each otiier in shape but are shifted along the frequency axis by the value a flie latter is named the temperature-shift factor. With war for an argument it becomes possible to plot temperature-invariant curves Re G (War) and lm G, (war). The temperature dependence of a is defined by the formula... 

The data are further analyzed mathematically, hi particular, it is of interest to establish retardation and relaxation time spectra that fit the measured data using Voigt or Maxwell models. Adding the temperature dependence of the data leads to the interesting observation that time and temperature effects are often coupled by the time-temperature superposition principle. Effects caused by an increase in temperature can also be produced by an increase in time scale of the experiment. The ratio of modulus to temperature, when plotted versus the logarithm of time for different temperatures,... 

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