Time-temperature superposition shift principle

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

T o = zero shear rate viscosity (steady-state viscosity at zero 

Tteat = relaxation time for tested specimen Tref = relaxation time for reference specimen ref = reference temperature, K Pref = reference density test = test temperature, K Ptest = test density 

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 

Ferry published a list of WLF Ci and C2 constants when Viscoelastic Properties of Polymers (2d ed., Wiley, New York, 1970). 

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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... 

Experimentally, one can use the time-temperature superposition (TTS) principle to extend the frequency range of data. It is often observed that rheological response measured at different temperatures is equivalent to one at the reference temperature To if one shifts the time (or frequency) appropriately. Sometimes, the stress has also to be shifted. For example, the complex relaxation modulus of theologically simple polymers defined as G (co) = G (co) +tG"(co), measured at different temperatures, obe3ts... 

The molecular theoiy predicts strong temperature dependence of the relaxation characteristics 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 the relaxation spectrum as a whole shifts in a self-similar... 

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. 

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. 

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 peak of the dielectric loss of this process reflects its viscoelastic nature by obeying the time-temperature superposition principle, wherein the peak is shifted to higher temperatures for shorter times (higher frequencies) and vice versa. This process has been described by the Havriliak-Negami empirical formula [106, 108]... 

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... 

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 ... 

The Time-Temperature Superposition Principle. For viscoelastic materials, the time-temperature superposition principle states that time and temperature are equivalent to the extent that data at one temperature can be superimposed upon data at another temperature by shifting the curves horizontally along the log time or log frequency axis. This is illustrated in Figure 8. While the relaxation modulus is illustrated (Young s modulus determined in the relaxation mode), any modulus or compliance measure may be substituted. 

It must be noted that changes in density, as well as changes in temperature for higher temperatures, result in vertical shifts. Usually, these are modest in size, however, compared to the horizonal shifts. It should also be noted that the WLF equation, above, is a corollary of the time-temperature superposition principle. 

The inherent difficulty in the measurement of the complex dynamic moduli of viscoelastic materials is emphasized by the results of this paper. The agreement among the shifted modulus data as measured by different systems is limited by several difficulties (1) measurement inaccuracies of the instruments, (2) differences in the data reduction techniques used to apply the time-temperature superposition principle and propagation of shift curve errors and, (3) nonuniformity of the test samples. 

However, for thermorheologically simple materials, that is, for those materials for which the time-temperature superposition principle holds, the mechanical properties data can be shifted parallel to the time or frequency axis. This fact suggests an additional hypothesis that can be very useful in solving some specific thermoviscoelastic problems. According to this hypothesis, the net effect of temperature in the response must be equivalent to a variation in the rates of creep or relaxation of the material. Thus for T > Tq the process occurs at a higher rate than at Tq. 

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. 

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... 

Typical friction force data obtained at a constant load are shown in Fig. 4.17 (left). In these experiments, the applied load should be limited to <1 nN to ensure that the surface of the film to a depth of <2 nm is probed and that wear of the glassy PMMA film can be excluded. A clear maximum is observed that shifts to higher velocities with increasing temperature, in accordance with the time—temperature superposition principle [36]. From the master curve of friction force vs. velocity data for PMMA (shifted by aT to the reference temperature of 25°C) an activation energy of 35 kJ/mol can be estimated. 

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... 

Figure 1. Linear dynamic oscillatory shear response of the 50K PBA based Si02 hybrid sample. The data collected at temperatures between 30 and 80 °C were reduced to a single master curve using the principle of time-temperature superpositioning. The horizontal frequency shift factors (af were similar to that for the pure PBA homopolymer.

These data can be used to show that time and temperature effects are often coupled for relaxation phenomena (the time/temperature superposition principle [13]). Effects due to an temperature increase can thus also be obtained by an increase of the experimental time scale. Hence, the curves in Figure 5.8 were shifted along the frequency-axis while the curve measured at 18°C was chosen as reference temperature. 

The time-temperature superposition principle, t-T, has been a cornerstone of viscoelastometry. It has been invariably used to determine the viscoelastic properties of materials over the required 10 to 15 decades of reduced frequency, COaj, [Ferry, 1980]. Measuring the rheological properties at several levels of temperature, T, over the experimentally accessible frequency range (usually two to four decades wide), then using the t-T shifting, made it possible to constmct the complete isothermal function. 

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