Time-temperature superposition shift factor

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

Time-temperature superposition [10] increases the accessible frequency window of the linear viscoelastic experiments. It applies to stable material states where the extent of reaction is fixed ( stopped samples ). Winter and Chambon [6] and Izuka et al. [121] showed that the relaxation exponent n is independent of temperature and that the front factor (gel stiffness) shifts with temperature... 

PPG (at higher temperatures) behaves like a typical pseudoplastic non-Newtonian fluid. The activation energy of the viscosity in dependence of shear rate (284-2846 Hz) and Mn was detected using a capillary rheometer in the temperature range of 150-180°C at 3.0-5.5 kJ/mol (28,900 Da) and 12-13 kJ/mol (117,700 Da) [15]. The temperature-dependent viscosity for a PPG of 46 kDa between 70 and 170°G was also determined by DMA (torsion mode). A master curve was constructed using the time-temperature superposition principle [62] at a reference temperature of 150°G (Fig. 5) (Borchardt and Luinstra, unpublished data). A plateau for G was not observed for this molecular weight. The temperature-dependent shift factors ax were used to determine the Arrhenius activation energy of about 25 kJ/mol (Borchardt and Luinstra, unpublished data). 

Apply time-temperature superposition principles to polymer moduli and calculate shift factors. 

Some applications require the material to remain under constant stress for years, yet it is often not reasonable to conduct such extended time measurements. One approach which circumvents this employs time-temperature superposition. Measurements are obtained over a shorter time span at differing temperatures. A master curve of C as a function of a reduced time tl a where a is a shift factor, is generated, and this allows the results to be extended to longer times. The shift factor is obtained by employing the Williams, Landel, and Ferry (WLF) relationship... 

Fig. Z4 (a) Temperature ramp at a frequency a> = lOrads (strain amplitude A = 2%) for a nearly symmetric PEP-PEE diblock with Mn = 8.1 X 104gmol l, heating from the lamellar phase into the disordered phase. The order-disorder transition occurs at 291 1 °C, the grey band indicates the experimental uncertainty on the ODT (Rosedale and Bates 1990). (b) Dynamic elastic shear modulus as a function of reduced frequency (here aT is the time-temperature superposition shift factor) for a nearly symmetric PEP-PEE diblock with Mn = 5.0 X 1O g mol A Shift factors were determined by concurrently superimposing G and G"for w > and w > " respectively. The filled and open symbols correspond to the ordered and disordered states respectively. The temperature dependence of G (m < oi c) for 96 < T/°C 135 derives from the effects of composition fluctuations in the disordered state (Rosedale and Bates 1990). (c) G vs. G"for a PS-PI diblock with /PS = 0.83 (forming a BCC phase) (O) 110°C (A) 115°C ( ) 120°C (V) 125°C ( ) 130°C (A) 135°C ( ) 140°C ( ) 145°C. The ODT occurs at about 130°C (Han et at. 1995).

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). 

From time-temperature superposition, the shift factors aT can be obtained by Eqs. 3.12 and 3.13. 

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...

In an earlier section, we have shown that the viscoelastic behavior of homogeneous block copolymers can be treated by the modified Rouse-Bueche-Zimm model. In addition, the Time-Temperature Superposition Principle has also been found to be valid for these systems. However, if the block copolymer shows microphase separation, these conclusions no longer apply. The basic tenet of the Time-Temperature Superposition Principle is valid only if all of the relaxation mechanisms are affected by temperature in the same manner. Materials obeying this Principle are said to be thermorheologically simple. In other words, relaxation times at one temperature are related to the corresponding relaxation times at a reference temperature by a constant ratio (the shift factor). For... 

Detailed analysis of the isothermal dynamic mechanical data obtained as a function of frequency on the Rheometrics apparatus lends strong support to the tentative conclusions outlined above. It is important to note that heterophase (21) polymer systems are now known to be thermo-rheologically complex (22,23,24,25), resulting in the inapplicability of traditional time-temperature superposition (26) to isothermal sets of viscoelastic data limitations on the time or frequency range of the data may lead to the appearance of successful superposition in some ranges of temperature (25), but the approximate shift factors (26) thus obtained show clearly the transfer viscoelastic response... 

The calculated shift factors for the 75/25 and 50/50 blends in the low temperature region (below 100°C) are close to the empirical shift factors for the pure PST phase. Above 140°C, a WLF-type behavior is found but with important deviations from PC. In between, the shift factors are time and temperature dependent. For the 25/75 blend (Figure 10c), no time dependence of log aT is found because time-temperature superposition is valid over the whole temperature domain. The relaxation behavior of this blend is completely dominated by the PST phase. The good agreement between the calculated and empirical values of the shift factors confirms again the validity of the mechanical model. 

Table I shows the values of these activation parameters for the materials tested. A time—temperature superposition shift factor (A) can be calculated from Equation 2 as follows ...

Tobol sky and co-workers who also modified it to account for proportionality of modulus to absolute temperature (3). This has the effect of creating a slight vertical shift in the data. Ferry further modified the time-temperature superposition to account for changes in density at different temperatures which has the effect of creating an additional vertical shift factor (4). The effect of the temperature-density ratio on modulus is frequently ignored, however, since it is commonly nearly unity. 

The flat appearance of the E" curve is due to the compressed nature of this particular nomograph scale. Both functions appear to fit equally well and therefore satisfy the criteria of curve shape and shift factor consistency for using the reduced variable time-temperature superposition. Additionally, the criterion of reasonable values for a-j- is satisfied by virtue of using the "universal" WLF equation. 

The utility of empirically determined WLF equations was investigated using DMA data obtained on the PVC acoustical damping material. Using a separate software package (available from DuPont Intruments), E, E" and tan 8 were empirically fit using the time-temperature superposition procedure. A reference temperature is first determined by the computer software. The data are then shifted manually and the WLF equation is fit to the resulting temperature shift factors. Values for and calcu-... 

Lopes da Silva et al. (1994) found that the fiequeney-temperature superposition, analogous to time-temperature superposition in transient rheologieal experiments, was applieable to a 1 % locustbean (LB) gum dispersion so that master eurves at To = 25°C were obtained for G and G" (Figure 3-38). In eontiast, smooth master eurves could not be obtained for G and G" values of 3.5% high-methoxyl pectin dispersions either separately or for both simultaneously (Figure 3-39). The discrepancies were higher for a 3.5% low-methoxyl pectin dispersion. It was concluded that the time-temperature superposition technique was not applicable to the pectin dispersions due to their aggregated structure. For the studied samples, the vertical shift factor for the moduli (Topo/ T P) had a small effect on the master curve (Lopes da Silva et al., 1994). 

LM pectin/calcium gels (Cp = 4-8.5 g/L R = 0.3-0.5) 11-86 Temperature dependence of the shift factor required for time-temperature superposition of mechanical spectra (5-40=C). Gamier (1992) o a... 

Because t] data are obtained at low strains with minimal alteration of the STD structure, they provide unique opportunities for studying applicable models. Further, empirically obtained frequency shift factor (Ferry, 1980) has been used successfully in time-temperature superposition studies on food polymer dispersions (Lopes da Silva etal., 1994), and the applicability ofsimilar, if not identical, scaling of frequency was explored for STDs (Yang and Rao, 1998a). 

An attempt will now be made to determine the activation energies of the two processes, since the identification of the mechanism is based primarily on this factor a study of all the stress relaxation curves has revealed that within experimental error time-temperature superposition is valid above a modidus of 10 dynes/cm , and that the shift factors are of the WLF form. Therefore, we can determine the constants of the WLF equation which were listed in Table 3 and also the corresponding activation energies. The values of the activation energies for the process which is governed by these shift factors, calculated for T —Tg + 30, ranges from 150 to 230 Kcal. This mechanism will, for the moment, be called the "first mechanism. 

The second important consequence of the relaxation times of all modes having the same temperature dependence is the expectation that it should -bp possible to superimpose linear viscoelastic data taken at different temperatures. This is commonly known as the time-temperature superposition principle. Stress relaxation modulus data at any given temperature Tcan be superimposed on data at a reference temperature Tq using a time scale multiplicative shift factor uj- and a much smaller modulus scale multiplicative shift factor hf. 

Time-temperature superposition also applies to other linear viscoelastic data, with the same shift factors. Two examples are the complex modulus in oscillatory shear,... 

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