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. 

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

Thus (he time scale / at /, divided by an is equivalent to the scale at On a log scale, log a, is thus the horizontal shift factor required for superposition. An important consequence of equation (22) is that a, or log (ii is the same for a given polymer (or solution) no matter what experiment is being employed. Thai is. creep and stress-relaxation curves are shifted by the same amount. 

Burnay [14] has developed a predictive model, which is based on the use of the superposition technique to determine thermal and dose rate shift factors relative to a master curve of compression set of a rubber seal versus time. The relation between the shift factors and environmental parameters of temperature and dose rate are given by ... 

Fig. 3.14. The data is for a very broad range of times and temperatures. The superposition principle is based on the observation that time (rate of change of strain, or strain rate) is inversely proportional to the temperature effect in most polymers. That is, an equivalent viscoelastic response occurs at a high temperature and normal measurement times and at a lower temperature and longer times. The individual responses can be shifted using the WLF equation to produce a modulus-time master curve at a specified temperature, as shown in Fig. 3.15. The WLF equation is as shown by Eq. 3.31 for shifting the viscosity. The method works for semicrystalline polymers. It works for amorphous polymers at temperatures (T) greater than Tg + 100 °C. Shifting the stress relaxation modulus using the shift factor a, works in a similar manner.

Fig. 4.5 Scaling representation of the spin-echo data at Q nax- Different symbols correspond to different temperatures. Solid line is a KWW description (Eq. 4.8) of the master curve, a Polyurethane at Qmax=l-5 A L The shift factors have been obtained from the superposition of the NSE spectra. (Reprinted with permission from [127]. Copyright 2002 Elsevier), b Poly-(vinyl chloride) at Qmax=l-2 A L The shift factors have been obtained from dielectric spectroscopy. (Reprinted with permission from [129]. Copyright 2003 Springer, Berlin Heidelberg New York)...

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. 7. Value of the shift factor, aT, needed for superpositioning of bulk mechanical data as a function of temperature, 5>. The various compositions of the rubber-modified epoxies are indicated...

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). 

The pattern can be obtained from the polymer temperature or concentration variations in addition to the change of G°N. The relaxation function may be too complicated a mathematical expression ever to be calculated, nonetheless, it obeys a property of invariance which allows the superposition of all normalised relaxation curves to one another by adjusting a suitable factor to the time scale of each curve. The time shift factor is found to obey the equation... 

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... 

In the present case, all of our dynamic mechanical data could be reduced successfully into master curves using conventional shifting procedures. As an example, Figure 7 shows storage and loss-modulus master curves and demonstrates the good superposition obtained. In all cases, the shifting was not carried out empirically in order to obtain the best possible superposition instead the appropriate shift factors were calculated from the WLF equation (26) ... 

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. 

Additionally, it was noticed that the shift factor for superposition fit the following empirical equation known as the WLF equation (5.6)... 

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-... 

The procedure by which the nomograph is generated is not limited to the WLF equation. Since it is based on the reduced variable concept, any superposition equation that results in the calculation of a temperature shift factor may be used to calculate the needed data to create the master curve and subsequent nomograph. The software can easily be modified to calculate and display a master curve on some other superposition equation. 

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