Time-Temperature Superposition TTS

The principle of TTS lies in the equivalency of time (frequency) and temperature. Due to various limitations, one cannot carry out experiments at conditions such as at very low frequencies and very high temperatures or vice versa. TTS is used to obtain data at different conditions to save experimental time. The viscoelastic data of one temperature can be related to the higher or lower temperature using a shift factor a ) to the right side, or to the left side of the time axis using a reference temperature (T f). A fully overlapped curve can be obtained for any reference temperature this is called a master curve . It is also widely accepted that a minor vertical shift factor may also be applied to more accurately model master curves. 

The master curve in the form of stiffness versus frequency can be created by fitting the experimentally determined shift factors to a mathematical model. With a multifrequency measurement, frequencies beyond the measurable range of the DMA can be achieved by using the superposition method based on the Williams-Landel-Ferry (WLF) equation [60, 61]. For a temperature range above the T, it is generally 

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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... 

Figure 3.71. An example of time-temperature superposition (TTS) of rheological data. Adapted from Figure 11.1 (Ferry, 1980). Cop)right (1980). Reprinted with permission of John Wiley and Sons, Inc.

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... 

Zhang et al. have shown that the conductivity curves can be transposed to each other by means of time temperature superposition (TTS) [62]. Furthermore, an Arrhenius plot of the corresponding shift factors exhibits that the dynamic percolation is a thermally activated process. With regard to the considered polycarbonate-MWCNT example, the Arrhenius plot of the fit parameter 2/cFy leads to the same result (Figure 5.26(b)). Since Vy f xo 1 constant, this means the reaction rate depends only on the activation energy, which is approximately 115 kJ/mol for this system. 

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... 

Figure 4.29. Time-temperature superposition (TTS) master curve for 7 et = 30°C obtained using the data in Fig. 4.28. Individual compliance curves recorded at each temperature shifted along the log[time] axis starting from 30 °C up to 145 °C created the master curve, (courtesy of TA Instruments.)...

While there is a relationship between time and temperature, the theories of viscoelasticity (Ferry 1980) do not deal with the temperature dependence. However, there is an empirical relationship referred to as the time-temperature superposition (TTS) principle, which provides a useful, practical... 

Time-Temperature Superposition In order to predict the long-term creep behavior based on short-term creep measurements, it is generally assumed that the polymer does not change its structure with time, and consequently the time-temperature superposition (TTS) principle can be adopted. ITS has been used to obtain the master curves for creep compliance against time. According to TTS, the creep at a given temperature (To) is related to the creep at another temperature... 

The composite natm e of polyurethane elastomers strongly affects their linear viscoelastic properties. It is known that for most polymers, linear viscoelastic moduli (storage modulus, E u,T), and loss modulus, E" u,T)) are characterized by the so-called time-temperature superposition (TTS) (see, e.g. Ferry [74]). Such behavior can be understood if one assumes that E (and E") is always a function of the product ut T), where t(T) is effective relaxation time. 

Polymeric materials exhibit viscoelastic phenomena, which must be taken into account in designing the materials applications. For example, rubber in a tire receives stimuli over a wide frequency and temperature range from the road surface. In the case of bulk samples, the frequency and temperature can be converted mutually based on the time-temperature superposition (TTS) principle [72]. However, TTS is a kind of empirical rule and, consequently, an actual measurement method with a wide frequency and temperature range is necessary to precisely predict the properties of practical products. Various AFM-based conventional methods have been proposed to measure viscoelasticity such as lateral force microscopy (LFM) [73-75], force modulation (FM) [76-78], and contact resonance (CR) [79-81]. Even tapping mode can report energy-dissipative phenomena [44,82-84] and further offers loss tangent mapping [85,86]. 

Also, even when the data are obtainable only over one or two decades of the logarithmic frequency scale at any one time, the viscoelastic functions can be traced out over a much larger effective range by making measurements at different temperatures, and by applying time-temperature superposition (TTS) for flexible homopolymers (see Chapter 6). In many instances, the effect of an increase in temperature is nearly equivalent to an increase in time or a decrease in frequency, as molecular viscoelastic theories suggest (see Chapter 4). When properly applied, TTS yields plots in terms of reduced variables that can be used with considerable confldence to deduce the effect of molecular parameters, and also to predict viscoelastic behavior in regions of the time or frequency scale not experimentally readily accessible (see Chapters 4 and 6). 

In this section, we present the experimental observations of the rheological behavior of some selected miscible polymer blends. Although there are so many pairs of miscible polymers reported in the literature, the number of studies reported on the rheological behavior of miscible polymer blend systems is rather small. Nevertheless, with the limited space available here, it is not possible to present the rheological behavior of every miscible blend system reported in the literature. Before presenting the rheological behavior of some specific miscible polymer blend systems, we first discuss the circumstances under which application of time-temperature superposition (TTS) to miscible polymer blends is warranted. 

One may wonder why we have not applied time—temperature superposition (TTS) to the dynamic frequency sweep data given in Figure 8.7, to determine the of SI-9/9, as numerous research groups (Adams et al. 1994 Balsara et al. 1998 Bates 1984 Bates et al. 1990 Floudas et al. 1994, 1996a, 1996b Lin et al. 1994 Modi et al. 1999 Rosedale and Bates 1990 Rosedale et al. 1995 Schulz et al. 1996 Wang et al. 2002 Winey et al. 1994) have done. As discussed in Chapter 6, application of TTS to flexible homopolymers has been practiced by two methods (1) by empirically... 

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... 

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