Time-temperature superposition shift

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).

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 ...

Fig. 14 Time-temperature superposition shift factor in rheoiogy, ax. as a ftuiction of temperature, normalised to Tg, for the supramolecuiar poiymer blend 7 and 8 (diamonds) and linear polystyrene (squares). (Reprinted with permission from [75], copyright 2009 RSC)...

It can be observed that the term of (1 -x) describes the contribution of Tf. Here, Tg is the glass transition temperature. denotes the Vogel temperature, defined as (Tg - 50) (°C), tq corresponds to the reference relaxation time at Tg, and B is the loeal slope at Tg of the trace of the time-temperature superposition shift factor in the global WiUiam-Landel-Ferry (WLF) equation [53]. 

To calculate the time-temperature superposition shift factor [2], use... 

To calculate the time-temperature superposition shift factor by using the WLF (Williams-Landel-Ferry) equation for polymers at temperatures less than 100°C (232 F) above their Tg [2], use... 

The free volume concept has been successfully used for describing the glass transition phenomenon and the Pick s law of diffusion tor polymers. One of such successful examples is the Williams-Landel-Ferry(WLF) equation [72,73], which provides the relationship between the time-temperature superposition shift factor and temperature. The free volume is considered to be the holes resulting from the packing void or irregularity of polymeric molecules. This concept will be continuously used in this section for deriving the viscosity equation ol both polymer melts and polymer solutions. 

Time-temperature superposition (tTs) was carried out for these multi-temperature multi-frequency tests based on Williams-Landel-Ferry (WLF) relationship. It considers the equivalency of time and temperature in the context of free volume theory for an activated flow process in viscoelastic materials such as PET. It has been found the tTs holds for the whole temperature/frequenQr range. The master curve generated from tTs is shown in Figure 13 for the 10 wt. % bamboo-PET composite at the 25.0 C reference temperature. The time-temperature superposition shift factor follows Arrhenius temperature dqrendence according to the expression ... 

V = process line speed (m/min) used in the experiment to bond the felt layers, ax = time-temperature superposition shift further defined by the relationship ... 

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... 

Time-temperature superposition at the gel point does not let us distinguish between the vertical and the horizontal shift, since the spectra are given by... 

Figure 8 WLF time-temperature superposition applied to stress-relaxation data obtained at several temperatures to obtain a master curve. The master curve, made by shifting the data along the horizontal axis by amounts shown in the insert for r> is shown with circles on a line.

The time-temperature superposition, implying that the functional form does not appreciably depend on temperature (see e.g. [34, 111]). For instance, mechanical or rheological data corresponding to different temperatures can usually be superimposed if their time/frequency scales are shifted properly taking a given temperature Tr as reference. 

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. 

The time-temperature superposition principle has practical applications. Stress relaxation experiments are practical on a time scale of 10 to 10 seconds (10 to 10 hours), but stress relaxation data over much larger time periods, including fractions of a second for impacts and decades for creep, are necessary. Temperature is easily varied in stress relaxation experiments and, when used to shift experimental data over shorter time intervals, can provide a master curve over relatively large time intervals, as shown in Figure 5.65. The master curves for several crystalline and amorphous polymers are shown in Figure 5.66. 

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... 

In addition to the primary glass-rubber relaxation which follows the empirical shifts determined by Eq. (26), part of the recoverable compliance does not obey time-temperature superposition. The shortest time data at the lowest temperatures has a component which shifts according to the Arrhenius temperature dependence... 

To obtain as much information as possible on a material, an empirical technique known as time-temperature superposition (TTS) is sometimes performed. This technique is applicable to polymeric (primarily amorphous) materials and is achieved by performing frequency sweeps at temperatures that differ by a few degrees. Each frequency sweep can then be shifted using software routines to form a single curve called a master curve. The usual method involves horizontal shifting, but a vertical shift may be employed as well. This method will not... 

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