Time-temperature superposition procedure

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 utility of empirically determined WLF equations was investigated using DMA data obtained on the PVC acoustical damping material. Using a separate software package (available from DuPont Intruments), E, E" and tan 8 were empirically fit using the time-temperature superposition procedure. A reference temperature is first determined by the computer software. The data are then shifted manually and the WLF equation is fit to the resulting temperature shift factors. Values for and calcu-... 

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. 

This time-temperature superposition procedure has the effect of producing a single continuous curve of modulus values extending over many decades of log t at the reference temperature. A curve constructed in this way, as shown in Figure 3.20 (with a reference temperature 115°C), is known as the master curve. 

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. 

Procedure 3 Obtaining a Mechanical Spectrum or Fingerprint of a Sample H3.1.6 Special Procedure Time-Temperature Superposition H3.1.7... 

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. 

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... 

The procedure for such data extrapolation is not arbitrary. The time-temperature superposition principle may be expressed mathematically for a stress relaxation experiment as... 

Figure 12.3, the procedure of time-temperature superposition for obtaining a master curve can be performed without any difficulty. Thus the compliance curves at different temperature levels differ only by the time scale, that is D(log t, T) = D(log t — log Uj) = D[log t/aj)],... 

Fig. 26. Influence of the temperature on logar of 9.9 wt% PVC/DOP. (O) Measuroimits correspond-ing with the extrapoktion procedure (A) measurements correspond with the normal application of the time-temperature superposition pincipfe (see Fig. 24). The full line corresponds with the WLF-equation (reference tenq)erature 1°Q. Re oduced from Physical Networks, Polymers and Gels [Ref. 12] by the courtesy of Chapman Hall...

Viscoelastic functions depend on both temperature and time. For many polymers, the logarithmic plot of a viscoelastic function at the temperature T may be obtained from that at the temperature Tq by shifting the curve along the logarithmic time axis by the amount of log (T)- This procedure is called time-temperature superposition. The ability to superpose viscoelastic data is known as thermorheological simplicity. Thermorheological simplicity demands that all the molecular mechanisms involved in the relaxation process have the same temperature dependencies. 

The relation between q and t of Eq. (6.4) appears by plotting q S(q) as a function of (qt ). An easy procedure is to use for the abscissa log (qt ) a large (q, t) range. The superposition of the curves in the range corresponding to Eq. (7.10a) should work for any direction. For the most complete set of data (set I), Fig. 8 shows the accuracy of the test. First, samples relaxed at same temperatures were compared to avoid the uncertainty caused by the time-temperature superposition. The largest uncertainty then results from the form factor, but would not artificially improve the fit in both directions e.g., if an error reduces the form factor in the perpendicular direction, as would a supplementary relaxation, it would also do so in the parallel direction, which would then appear less relaxed. The result of this fit is ... 

When TTS was originally implemented, the distance that a curve must be shifted in order to transpose it from the temperature of observation T to the reference temperature Tr was determined by the simple empirical curvefitting procedure illustrated above. Pioneering research on this aspect of TTS was carried out by Leaderman (1941, 1943) and Tobolsky (1960), who also modified the procedure to account for the proportionality of modulus to absolute temperature. This has the effect of creating a slight vertical shift in the data. Williams et al. (1955) further modified time-temperature superposition to account for changes in density at different temperatures, which has the... 

Again, as for the section on time dependent yielding it is necessary to consider how to obtain data that can be useful in preventing failure over the lifetime of a structure that may be intended to last for 20 to 100 years. The only rational means for this is some type of accelerated testing such as that offered by the time-temperature or time-stress-superposition procedure. Also, it is reiterated that failure is a stochastic event and of necessity a reliable statistical analysis should be performed. 

---