Horizontal shift factor

Curves for the viscosity data, when displayed as a function of shear rate with temperature, show the same general shape with limiting viscosities at low shear rates and limiting slopes at high shear rates. These curves can be combined in a single master curve (for each asphalt) employing vertical and horizontal shift factors (77—79). Such data relate reduced viscosity (from the vertical shift) and reduced shear rate (from the horizontal shift). 

Thus (he time scale / at /, divided by an is equivalent to the scale at On a log scale, log a, is thus the horizontal shift factor required for superposition. An important consequence of equation (22) is that a, or log (ii is the same for a given polymer (or solution) no matter what experiment is being employed. Thai is. creep and stress-relaxation curves are shifted by the same amount. 

Master curves can often be made for crystalline as well as for amorphous polymers (33-38). The horizontal shift factor, however, will generally not correspond to a WLF shift factor. In addition, a vertical shift factor is generally required which, has a strong dependence on temperature (36-38). At least part of the vertical shift factor results from the change in... 

The plot of a master curve of the thermal variations of rio for various molecular weights and temperatmes (Fig. 23) shows that the expansion coefficient Uf can be considered as a constant in a wide range of temperatures. The vertical and horizontal shift factors respectively describe the mass dependence of the radius of gyration and temperature T . 

Figure 24 The horizontal shift factors of the master curve of Fig. 23 give the temperature T as a function of molecular weight (reference = 49.4° C).

The variations of are derived from the horizontal shift factor and can be expressed by (Fig. 24) ... 

These equations lead to the following expression for the horizontal shift factor (2,3) ... 

The variation of density of rock resulting from temperature variation is very small. Therefore it can be considered that the intrinsic variation of creep compliance itself is also small. Consequently the effect of vertical shift factor in eq. (17) can be neglected and only the horizontal shift factor plays a dominant role. 

Horizontal shift factors (a ) were determined empirically. No vertical shift factor (To/T) was Incorporated. The empirical shift factors were used with Arrhenlus-type [log = (. AH/2.303R)/ (1/T-l/To)] and HLF-type [log at - Ci (T-To)/(C2 + T-To)] equations. A plot of T-To/log at versus T-To yielded HLF constants and C< 2 resultant values corresponding to the glassy zone... 

For materials that obey the time-temperature superposition principle, the dynamic moduli can be rescaled using appropriate shift factors to obtain a master curve. The temperature dependence of the horizontal shift factor is usually described by the WLF equation. Explain how the WLF parameters (Zi, Oj. and can be determined from a set of log % vs. temperature data. 

The horizontal shift factor reflects the temperature dependence of relaxation time, and the vertical shift factor reflects the... 

Fig. 10.48 Horizontal shift factors for the pure components and the singlephase 20/80 PS/PVME blend at a reference temperature of 1 °C (Polios 1997)...

A typical example of the master curve [2] is shown in Figure 9.3 for HEUR (polyethylene oxide) end-capped with -C16H33. The reference temperature is chosen at 5 C. From the horizontal shift factor, the activation energy is found to be 67kJmol . From the high-frequency plateau of the storage modulus, the number of elastically effective chains is found as a function of the polymer concentration, which was already studied in Section 8.2 (Figure 8.10). 

Perhaps the earliest work in this vein concerned the time-moisture effects on the relaxation of PVA and nylon 6 films (Onogi et al. 1962). Relaxation data collected at distinct levels of relative humidity were coalesced through horizontal shifts to form a master relaxation curve as shown in Figs. 6.1 and 6.2. The horizontal shift factor function an is plotted vs. RH in Fig. 6.3. 

Fig. 6.1 Horizontal shift factors at various levels of relative humidity...

Fig. 6.3 Horizontal shift factors vs. relative humidity and absorbed water content for PVA nylon six films of two degrees of polymerization...

The factor is thus a horizontal shift factor that can be used to shift data taken at a temperature... 

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