Shift factors, logarithmic temperature

FIGURE 5.7 Logarithmic plot of the time scale shift factors against temperature differences. The atmosphere in which the measurements were made is either in air containing the moisture absorbed under ambient condition or in arough vacuum that is partially dry. Tco = —57, — 54, — 50, — 48°C, respectively, for Samples 11-A (air), 10-A (vacuum), 10-B (air), 10-B (vacuum). 

Fig. 14. Plot of shift factor (logarithm) vs T for polycarbonate, illustrating the change in behavior away from the WLF-type of temperature dependence as Tg is traversed. Also, because the glassy state is a nonequilibrium state, the material ages. The curves at 0.5, 2, and 16 h represent the results at these aging times. After Pesce et al (33).

Curves of stress (divided by absolute temperature) versus log time-to-break at various temperatures can be made to coincide by introducing the temperature-dependent shift factor flT. Application of the same shift factor causes the curves of the elongation at the break br versus the logarithm of time-to-break at various temperatures to coincide. A direct consequence is that all tensile strengths (divided by absolute temperature), when plotted against elongation at break, fall on a common failure envelope, independent of the temperature of testing. Fig. 13.84 shows the behaviour of Viton B elastomer. 

Figure 3. Master curve of indenter modulus of jacketing based on chlorosulfonatedpolyethylene at 155°C obtained by horizontal shifting of the data taken at the other temperatures (120-200°C) by log ar along the logarithmic time axis. The samples were aged in nitrogen. The temperature-dependence of the shift factor is shown in the insert figure. From Sandelin and Gedde (13) and with permission from Elsevier, UK.

Figure 5.7 Frequency dependences of the storage (Q) and loss (-h) moduli for poly(dimethylsilox-ane) (PDMS) samples whose reactions were quenched at the times indicated (see Fig. 5-6). The data are time-temperature-shifted to the reference temperature T gf of 34°C, and they are shifted additionally by an amount A on the logarithmic axis to keep the curves from overlapping. The vertical shift factors bf are given by p T sf)T d/ p T)T), where p is the density. (From Winter and Chambon 1986, with permission from the Journal of Rheology.)...

Figure 12.24 Double logarithmic plot of the aging time shift factors versus aging time, tg, for an epoxy glass aged at different temperatures below its Tg. T — Tg. (O) 30°C, (X) 24°C, ( ) 20.8°C (O) 10.3°C ( ) 6.TC. (From Ref. 30.)...

Similarly, the group of logarithmic curves of cohesion vs. time of the rock are obtained shown in Figure 3, by a series of triaxial compressive creep tests at seven temperatures from 20°C to 300°C. In the same way as mentioned above we can also obtain the main curve (Figure 4) at the reference temperature at 20°C as well as the corresponding shift factor parameters (Table 2). 

Because the time or frequency scales over which the compliance changes are very large, it is usual to use logarithmic scales for t or co. When this is done it is necessary to shift the curves for different temperatures by a constant amount log a or — log aj along the t or co axes, respectively, in order to get superposition with the curve for T. The quantity aj is therefore called the shift factor. Figure 7.14 shows data for a set of measurements before and after the shifts have been applied. The idea that the same effect can be produced on the compliances or moduli either by a change of temperature or by a change of time-scale is called time-temperature equivalence. 

The time scale shift factors aj that were determined in the reduction of the creep compliance curves to obtain the reduced curves shown in Figure 5.4 are presented in Figure 5.5. The logarithm of aj is plotted as a function of the reciprocal absolute temperature. The temperature dependence data can be fitted... 

FIC U RE 5.5 Logarithmic temperature shift factors log aj plotted as functions of the reciprocal absolute temperature T/K for the four indicated epoxy resins which were obtained from the reduction process used in producing the curves in Figure 5.8. 

A typical example of these measurements is the mechanical loss factor (for definition, see Section 11). Here a loss maximum for poly (cyclohexyl methacrylate) is observed at — 125°C when the frequency is 10 Hz (Figure 10-26). The maximum is shifted to higher temperatures when the frequency is increased. In addition, the reciprocal loss temperature depends linearly on the logarithm of the frequency (Figure 10-27). Studies on different chemical compounds show that this loss maximum is specific to the cyclohexyl group. The values for both poly(cyclohexyl methacrylate) and poly(cyclohexyl... 

The horizontal logarithmic time scale shifts that are required to superpose the data obtained at different temperature are the logarithms of the Ut shift factors. The Uj values thus reflect the principal temperature dependence of the viscoelastic process. It was possible to represent the time-scale temperature dependences of the three samples with a single VFTH Eq. (33) in which only one parameter T , which reflects the change in Tg, varies with the level of crosslinking. The fit achieved is shown in Fig. 12. The atmosphere in which the measurements were made is important since samples measured in air contain the moisture absorbed under ambient conditions, whereas those measured in rough vacuum (use about symbol - lO torr = 1.3 Pa) are at least partially... 

The WLF equation enables static glass-transition temperatures Tq and various dynamic glass-transition temperatures T to be interconverted. To do this, the deformation times for the various individual methods must be known (Table 10-8). The shift factor a, for the calculation is obtained as the difference between the logarithms of the deformation times. 

It should be noted that the shift factors that can be fitted to the WEE equation show positive curvature when plotted logarithmically against the temperature. Quite often, ogaj values obtained near and below Tg show negative curvature at low temperatures simply is an indication that the lower temperature measurements were made before the density of the material reached its equilibrium value. 

Shift factor n. The amount by which the logarithm of the modulus (or comphance) of a plastic, measured at temperature T (K) must be shifted along the time axis to bring it onto a single curve with the modulus measured at Tg, the glass-transition temperature the shift factor relationships is... 

---