Viscoelastic data, time-temperature superposition

Since we are interested in this chapter in analyzing the T- and P-dependences of polymer viscoelasticity, our emphasis is on dielectric relaxation results. We focus on the means to extrapolate data measured at low strain rates and ambient pressures to higher rates and pressures. The usual practice is to invoke the time-temperature superposition principle with a similar approach for extrapolation to elevated pressures [22]. The limitations of conventional t-T superpositioning will be discussed. A newly developed thermodynamic scaling procedure, based on consideration of the intermolecular repulsive potential, is presented. Applications and limitations of this scaling procedure are described. 

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... 

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. 

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. 

Time-temperature superposition was first suggested by H. Leaderman who discovered that creep data can be shifted on the horizontal time scale in order to extrapolate beyond the experimentally measured time frame (9-10). The procedure was shown to be valid for any of the viscoelastic functions measured within the linear viscoelastic range of the polymer. The time-temperature superposition procedure was first explicitly applied to experimental data by... 

The time-temperature superposition method can be also applied to viscosity data (Ferry, 1980). For any viscoelastic parameter, exact matching of the adjacent curves is an important criterion for the applicability of the method. In addition, when possible, the same values of oy must superpose all the viscoelastic parameters and the temperature dependence of ar should have a reasonable form based on experience. One advantage of the method is that the range of frequencies are extended beyond those available experimentally. The time-temperature method has been also referred to as thermorheological simplicity (Plazek, 1996). 

Linear Viscoelasticity of Unfractionated Samples. The BP6L and BP6H samples were found to give reproducible data at temperatures below 120°C if first exposed to 150°C for 5 minutes. After such a heat treatment measurements were made on these samples at T = 35, 41, 50, 60, 70, 80, 90, 101, and 120°C. The empirical time-temperature superposition principle (13) was found to be valid for BP6L between 60°C and 120°C and for BP6H between 40°C and 120°C, and was used to make master curves at a reference temperature of 101°C (Figs. 4 and 5). The modulus scale... 

Viscoelastic Master Curves. In order to evaluate whether a given material is suitable for a particular damping application we need to know its viscoelastic properties over a broad range of temperature and frequency. However, in most instances we can measure these quantities only over a limited range of temperature or frequency. The data are then extended to other temperatures and frequencies by using the time-temperature-superposition procedure (5, 6) to form viscoelastic master curves that correlate the data and extend its utility. 

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. 

Time-temperature superposition also applies to other linear viscoelastic data, with the same shift factors. Two examples are the complex modulus in oscillatory shear,... 

Validity of Eqs 7.81 and 7.82 was examined for mixtures of entangled, nearly monodispersed blends of poly(ethylene-a/r-propylene) with head-to-head PP [Gell et al., 1997]. The viscoelastic properties compared at constant distance from the glass transition temperature of each system were found to obey the time-temperature superposition principle. The data agreed better with the predictions of Eq 7.82 than Eq 7.81. However, for blends of linear and branched PE the relations 7.82 were found valid only when MW and rheological properties of the two components were similar [Groves et al., 1996]. 

Fortunately for linear amorphous polymers, modulus is a function of time and temperature only (not of load history). Modulus-time and modulus-temperature curves for these polymers have identieal shapes they show the same regions of viscoelastic behavior, and in each region the modulus values vary only within an order of magnitude. Thus, it is reasonable to assume from such similarity in behavior that time and temperature have an equivalent effect on modulus. Such indeed has been found to be the case. Viscoelastic properties of linear amorphous polymers show time-temperature equivalence. This constitutes the basis for the time-temperature superposition principle. The equivalence of time and temperature permits the extrapolation of short-term test data to several decades of time by carrying out experiments at different temperatures. 

Time-temperature superposition is applicable to a wide variety of viscoelastic response tests, as are creep and stress relaxation. We illustrate the principle by considering stress relaxation test data. As a result of time-temperature correspondence, relaxation curves obtained at different temperatures can be superimposed on data at a reference temperature by horizontal shifts along the time scale. This generates a simple relaxation curve outside a time range easily accessible in laboratory experiments. This is illustrated in Figure 14.13 for polyisobutylene. Here, the reference temperature has been chosen arbitrarily to be 25°C. Data obtained at temperature above 25°C are shifted to the right, while those obtained below 25°C are shifted to the left. 

This time-temperature superposition of linear viscoelastic data means that all the retardation times t, of the linear viscoelastic model have a common temperature shift factor a(T)... 

The most common means to extend the frequency scale is to invoke time-temperature superpositioning (Ferry, 1980). If all motions of a polymer contributing to a particular viscoelastic response are affected the same by temperature, then changes in temperature only alter the overall time scale such a material is thermorheologically simple. Thermorheological simplicity means conformance to the time-temperature superposition principle, whereby lower and higher strain rate data can be obtained from measurements at higher and lower temperatures, respectively. 

If the modulus data of a material satisfy Equation (3.27), this material is referred to as thermorheologically simple and obeys the (viscoelastic) time-temperature superposition. 

For the moduli data, the time-temperature superposition fails at intermediate (0 between the segmental and global relaxation processes because these processes exhibit different Ojq at low T - (see, e.g., Adachi and Kotaka, 1993 Inoue et al., 1991, 1996 Kremer and Schonhals, 2003). (This failure is not well resolved in the compressed scale of the plots shown in Figure 3.3.) The superposition works separately at high and low ca where the viscoelastic data are dominated by one of these processes. In contrast, the dielectric data satisfy the superposition in the entire range of co because those data detect just the segmental relaxation process, although it fails in a close vicinity of... 

Fig. 3.4. With a multi-frequency measurement, frequencies beyond the measurable range of the DMA can be achieved by using the superposition method. Employing the Williams-Landel-Ferry (WLF) equation, and with a treatment of the data, designated as the method of reduced variables or time-temperature superposition (TTS) it is possible to overcome the difficulty of extrapolating limited laboratory tests at shorter times to longer-term, more real world conditions. The underlying bases for TTS are that the processes involved in molecular relaxation or rearrangements in viscoelastic materials occur at accelerated rates at higher temperatures and that there is a direct equivalency between time (the frequency of the measurement) and temperature.

Non-linear mechanical properties were observed for rubber eomposites and referred to as the Payne effect. The Payne effeet was interpreted as due to filler agglomeration where the filler clusters formed eontained adsorbed rubber. The occluded rubber molecules within filler elusters eould not eontribute to overall elastic properties. The composites behaved similarly to rubber composites with higher filler loading. Uniform and stable filler dispersion is required for rubber composites to exhibit linear viscoelastic behaviour. Payne performed dielectric measurements on SBR vulcanizates containing silica or carbon black. The dielectric data were used to construct time-temperature superposition master curves. The reference temperature increased with crosslinking but not significantly with filler. Comparison of dynamic mechanical and dielectric results for the SBR blended with NR was made and interpreted. ... 

Yu et al. (2011) studied rheology and phase separation of polymer blends with weak dynamic asymmetry ((poly(Me methacrylate)/poly(styrene-co-maleic anhydride)). They showed that the failure of methods, such as the time-temperature superposition principle in isothermal experiments or the deviation of the storage modulus from the apparent extrapolation of modulus in the miscible regime in non-isothermal tests, to predict the binodal temperature is not always applicable in systems with weak dynamic asymmetry. Therefore, they proposed a rheological model, which is an integration of the double reptation model and the selfconcentration model to describe the linear viscoelasticity of miscible blends. Then, the deviatirMi of experimental data from the model predictions for miscible... 

Williams-Landel-Ferry equation that relates the value of the shift factor, ax (associated with time-temperature superposition of viscoelastic data), required to bring log-modulus (or log-compliance) vs. time or frequency curves measured at different temperatures onto a master curve at a particular reference temperature. To, usually taken at 50 °C above the glass transition temperature (To = Tg + 50 °C) ... 

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