Time-temperature-superposition master curves

To get accurate distributions of relaxation or retardation times, the expetimcntal data should cover about 10 or 15 decades of time. It is impossible to get experimental data covering such a great range of times at one temperature from a single type of experiment, such as creep or stress relaxation-t Therefore, master curves (discussed later) have been developed that cover the required time scales by combining data at different temperatures through the use of time-temperature superposition principles. 

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.

Figure 3.15 Modulus-time master curve at 115°C based on time-temperature superposition of the PMMA resin data shown in Fig. 3.14 (Fig. 8.13 in Rodriguez [1]). The T for PMMA resin is 105°C...

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the time-temperature superposition principle [62] at a reference temperature of 150°G (Fig. 5) (Borchardt and Luinstra, unpublished data).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shift factors ax were used to determine the Arrhenius activation energy of about 25 kJ/mol (Borchardt and Luinstra, unpublished data). 

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... 

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... 

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). 

FIG. 13.48 Small-strain tensile creep of rigid PVC. Left short-time tests (t < 1000 s) at a te of 2 h after quenches from 90 °C to various temperatures (f/fe < 0.13). The master curve at 20 °C was obtained by time-temperature superposition (compare Section 13.4.8) the dashed curves indicate the master curves at other temperatures. Right, long-term tests (t = 2 x 106 s, fe = 1/2 h, t/te = 1100). The dashed lines are the master curves at 20 and 40 °C for a te of 1/2 h they were derived from the left-hand diagram. From Struik (1977,1978). Courtesy of the author and of Elsevier Science Publishers. 

FIG. 13.56 Use of the time-temperature superposition illustrated with polyisobutylene data. The reference temperature of the master curve is 25 °C. From Eisele (1990), as reconstructed from Castiff and Tobolsky (1955,1956). Courtesy Springer Verlag. 

Fig. 21. Storage modulus (G ) for PCL based silicate nanocomposites. Silicate loadings are indicated by percentual values in the figure. Master-curves were obtained by application of time-temperature superposition and shifted to T0=55 °C. From Ref. [54].

Almost always the data from the apparatus above is analyzed by using the time-temperature superposition principle to form a master curve over a wide frequency range at a selected reference temperature. The basis for this procedure is that for thermorheologically simple materials the effect of a change in temperature on... 

A Fortran based program has been developed for the IBM XT (or compatible) that uses the principles of time-temperature superposition and the WLF equation to generate master curves of DMA data. 

The acoustical properties for all samples were evaluated by using either a Toyo DDV-II Rheovibron viscoelastometer at 110 Hz, or a string apparatus developed at NRL-USRD, (8). In this latter instrument, dynamic Young s moduli and loss tangent were measured in the frequency range of 1-10 kHz, and master curves were obtained by using the time-temperature superposition technique, (9). 

The similarities of the isothermal curves permit the construction of a master curve (thick dashed curve in Figure 13.4) for any reference temperature To by the well-known time-temperature superposition principle [30,36]. The superposed master curve expands the experimentally accessible a> window for these examples by about three decades relative to that measured at any single temperature. This wider window is critical to understanding fabrication performance. 

Figure 17 shows master curves of the orientational relaxation of short deuterated chains of various molecialar weights present at 20 wt.-% in a matrix of molecular weight 2x10 g/mol (PS 2000). The master curves have been drawn at a reference temperature of Tg + 9 °C using time-temperature superposition. 

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