Time Horizontal shift factor

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. 

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... 

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. 

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... 

Stress relaxation master curve. For the poly-a-methylstyrene stress relaxation data in Fig. 1.33 [8], create a master creep curve at Tg (204°C). Identify the glassy, rubbery, viscous and viscoelastic regions of the master curve. Identify each region with a spring-dashpot diagram. Develop a plot of the shift factor, log (ax) versus T, used to create your master curve log (ot) is the horizontal distance that the curve at temperature T was slid to coincide with the master curve. What is the relaxation time of the polymer at the glass transition temperature ... 

Horizontal shift distances at different points between the relaxation isotherms were measured. For PC (7), PST (8), and the 25/75 blend, the shift factors are independent of time, and complete master curves can be obtained by simple tj nf g laxation isotherms along... 

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.

The shift factors are usually obtained by empirical methods that involve the horizontal translation of the isotherm representing the reduced viscoelastic functions in the time or frequency domains, in double logarithmic plots with respect to the reference isotherm. However, analysis of the components of the complex relaxation moduli in the terminal region ( 0) permits... 

Figure 1. Linear dynamic oscillatory shear response of the 50K PBA based Si02 hybrid sample. The data collected at temperatures between 30 and 80 °C were reduced to a single master curve using the principle of time-temperature superpositioning. The horizontal frequency shift factors (af were similar to that for the pure PBA homopolymer.

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.

The effect of nonlinearity increases gradually with increasing stress. It is seen from Figure 12.7 that the compliance curves at different stress levels are displaced relative to each other along the log time axis. These curves were shifted horizontally to produce the master curve consisting of several different symbols (see Figure 12.7) in Figure 12.8 [36]. The stress dependence of the stress shift factor is shown in an insert. The reference stress for the master curve is 30 MPa. 

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... 

FIGURE 28 Comparison of the [Jp t) - /J curves as a function of the reduced retardation time, t/a-T, of fully cured Epon 828/DDS (O), 1001/DDS (0), 1004/DDS (A), and 1007/DDS (V). The shift factors and reference temperature for Epon 828/DDS are identical to the values used in Fig. 8. For the other three elastomers, horizontal shifts of -0.72, -1.18, and -1.5 have been applied to superimpose the data in the short time regime associated with segmental relaxation. 

If a series of stress relaxation curves is obtained at varying temperatures, it is found that these curves can be superimposed by horizontal shifts to produce a master curve. This demonstrates an important feature in polymer behavior the concept of time-temperature equivalence. In essence, a polymer at temperatures below room temperature wiU behave as if it were tested at a higher rate at room temperature. This principle can be applied to predict material behavior under testing rates or times that are not experimentally accessible through the use of shift factors oT) and the equation below ... 

Time-temperature equivalence in its simplest form implies that the viscoelastic behaviour at one temperature can be related to that at another temperature by a change in the time-scale only. Consider the idealized double logarithmic plots of creep compliance versus time shown in Figure 6.7(a). The compliances at temperatures T and T2 can be superimposed exactly by a horizontal displacement log at, where at is called the shift factor. Similarly (Figure 6.7(b)), in dynamic... 

The horizontal shift on a logarithmic time-scale is shown in Figure 6.14. Remarkably, Williams, Landel and Ferry [7] found an approximately identical shift factor-temperature relation for all amorphous polymers, which could be expressed as... 

Fig. 4.20. Illustration of the time-temperature shift (eqn (4.72)). The shear compliance curves at T and 7q when plotted against log t are simply displaced horizontally by log a7- a is the shift factor (eqn (4.68)). The small temperature dependence of J J and is neglected here.

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