Time Vertical shift factor

Tobol sky and co-workers who also modified it to account for proportionality of modulus to absolute temperature (3). This has the effect of creating a slight vertical shift in the data. Ferry further modified the time-temperature superposition to account for changes in density at different temperatures which has the effect of creating an additional vertical shift factor (4). The effect of the temperature-density ratio on modulus is frequently ignored, however, since it is commonly nearly unity. 

Lopes da Silva et al. (1994) found that the fiequeney-temperature superposition, analogous to time-temperature superposition in transient rheologieal experiments, was applieable to a 1 % locustbean (LB) gum dispersion so that master eurves at To = 25°C were obtained for G and G" (Figure 3-38). In eontiast, smooth master eurves could not be obtained for G and G" values of 3.5% high-methoxyl pectin dispersions either separately or for both simultaneously (Figure 3-39). The discrepancies were higher for a 3.5% low-methoxyl pectin dispersion. It was concluded that the time-temperature superposition technique was not applicable to the pectin dispersions due to their aggregated structure. For the studied samples, the vertical shift factor for the moduli (Topo/ T P) had a small effect on the master curve (Lopes da Silva et al., 1994). 

Therefore, Or = 0.0012. This means that the longest relaxation time r of the polyisoprene at 100°C is 0.0012 times its value at 25°C. The viscosity tj changes roughly in proportion to the relaxation time if the small vertical shift factor is neglected. Thus, ... 

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 6.7 Atactic polypropylene segmental relaxation times (solid symbols) from mechanical spectroscopy, dynamic light scattering, dielectric relaxation, and NMR, along with the global time-temperature shift factors (open symbols) from dynamic mechanical spectroscopy, creep compliance, and viscosity. Vertical shifts were applied to superpose the data (Roland et al., 2001). 

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

Vertical shifts are seen as a temperature dqiradait zero time shear modulus Gq the vertical shift factor is hr = GtP /G,fJ). The modulus G is calculated ftxnn the torque 1 assuming that die stress and strain ate both linear ftmctitMis of the t linder radius, as in Equation (1). 

Figure 3. Tonperature shift ftu tors, log as a function of temperature. Table II. Vertical shift factors for time - temperature and time - strain superposition...

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. 

A more refined analysis reveals that, if the correlation times precisely superimpose on the WLF curve for PB and Stereon 704 and 705, their values are smaller in Solprene 1204. This change in mobility can be interpreted by an additional effect of microstructure upon the T (Tg) vertical shift factor. Solprene 1204 differs from PB and its other copolymers by its microstructure (Table 1). The change of the elastomer phase in Solprene 1204, consisting of an increase of the vinyl conformations to the detriment of the 1,4 cis or trans conformations would induce a diminution of T (Tg). In other words, it appears that the vertical shift factor T (Tg relative to the WLF equation is slightly different in Solprene 1204 from its value in PB and Stereon 704 and 705. A further shift of the experimental correlation times for the Solprene 1204 matrix indicates in Fig, 6 that the temperature dependence in PB and its copolymers is well-fit by the WLF equation. 

This implies that if the vertical shift factor is neglected, the time shift factor can be obtained as a function of temperature by measuring the zero-shear viscosity at several temperatures. 

The study that produced the data shown in Fig. 10.2 [2] included several other solutions, and it was found that molecular weight and concentration had little effect on the damping function for cM around 5 lO. Todemonstrate the degree of time-strain separability, the data ofFig. 10.2 are replotted in Fig. 10.5 as relaxation modulus divided by the vertical shift factor required to superpose them, i.e., The superposibility is excellent for times longer than a... 

Thus all the different temperature related data in Fig. 2.58 could be shifted to a single master curve at the reference temperature (7 ). Alternatively if the properties are known at Tref then it is possible to determine the property at any desired temperature. It is important to note that the shift factor cannot be applied to a single value of modulus. This is because the shift factor is on the horizontal time-scale, not the vertical, modulus scale. If a single value of modulus 7, is known as well as the shift factor ar it is not possible to... 

Fig. 11.15 The apparent time-shift factor, /rc, as a function of the molecular weight (MW) of the low MW component (denoted by Y) in the binary blends F80/Y = 75/25 ( ) F80/Y = 50/50 (o) F35/Y = 75/25 (a) and F35/Y = 50/50 (a). The horizontal lines on the vertical axis indicate the theoretical rf, /tc values calculated from Eki- (11.10) as the limiting values for the four series of blend samples.

Fig. 19. Temperature dependence of the shift factors of the viscosity (T), terminal dispersion ( ), and softening dispersion (0) of app from Ref. 73. The temperature dependence of the local segmental relaxation time determined by dynamic light scattering ( ) (30) and by dynamic mechanical relaxation (o) (74). The two solid lines are separate fits to the terminal shift factor and local segmental relaxation by the Vogel-Fulcher-Tammann-Hesse equation. The uppermost dashed line is the global relaxation time tr, deduced from nmr relaxation data (75). The dashed curve in the middle is tr after a vertical shift indicated by the arrow to line up with the shift factor of viscosity (73). The lowest dashed curve is the local segmental relaxation time tgeg deduced from nmr relaxation data (75).

Fig. 2.29 ( ). In the same figure are shown the Rouse relaxation times, tr (a), and the sub-Rouse relaxation times, t r (o), obtained from the peaks of tan (5 in Fig. 2.28. The dashed and dotted curves drawn through them are fits to tr and Tsr data produced by using the WLF equation. The two vertical arrows at T = Ta,sR and T = rsR,R divide the temperature into three regimes, I, II, and III. In regime I, the mechanical responses obtained by measurements of creep compliance [210] or stress relaxation [2] are mainly in the range /g < J(t) < 10 Pa and hence contributed by the local segmental relaxation. Thus it is appropriate to fit the creep data in regime I to Eq. (2.33) with 1 — = 0.55 to determine t . Shift factors aj used for time-...

Fig. 2.33. A comparison of the retardation spectra L of a high molecular weight PS (filled triangles), a solution of 25% PS in TCP (open squares) and PIB (filled circles). The shift factors are arranged such that the maximum of the first peak occurs at the same reduced frequency for all three samples. Downward vertical shifts by 0.869 and 1.39 of logio L have been applied to data for PS and the 25% PS solution, respectively, in order to make all data have about the same height at the first maximum. The disparity in width of the softening dispersion of bulk PS and PIB is clear. The small peak near the bottom (dashed line) is the contribution to L from the local segmental motion in bulk PS. The inset shows isothermal tan 8 data of PIB in the softening region at -66.9 °C, and tan 8 of the solution of 25% PS in TCP obtained from a reduced recoverable-compliance curve after applying time-temperature superposition to the limited isothermal data.

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