Temperature-shift factor

If the values of shift factor logjo ut, obtained as above, are plotted against test temperature T, a smooth curve is obtained. Williams, Landel and Ferry (1955) showed that the same shift factor-temperature relationship was obtained from the experimental shifting of results from a large number of amorphous polymers. The empirical relationship thus obtained is known as the WLF equation, and can be used in one of the two forms ... 

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... 

The remarkable observation, which was established largely by the work of Williams, Lan-del and Ferry, is that for all amorphous polymers this shift factor-temperature relationship is approximately identical. 

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

We shall presently examine the physical significance of the shift factors, since they quantitatively embody the time-temperature equivalence principle. For the present, however, we shall regard these as purely empirical parameters. The following Ust enumerates some pertinent properties of a ... 

From the data in Fig. 4.8b, estimate the shift factors required to displace the data at 0 = 0.5 (consider only this point) so that all runs superimpose on the experiment conducted at 128 C at 0 = 0.5. Either a ruler or proportional dividers can be used to measure displacements. Criticize or defend the following proposition Whether a buffered aqueous solution of H2O2 and 1. containing small amounts of S2O3 and starch, appears blue or colorless depends on both the time and the temperature. This standard general chemistry experiment could be used to demonstrate the equivalency of time and temperature. The pertinent reactions for the iodine clock are... 

The time-temperature superpositioning principle was applied f to the maximum in dielectric loss factors measured on poly(vinyl acetate). Data collected at different temperatures were shifted to match at Tg = 28 C. The shift factors for the frequency (in hertz) at the maximum were found to obey the WLF equation in the following form log co + 6.9 = [ 19.6(T -28)]/[42 (T - 28)]. Estimate the fractional free volume at Tg and a. for the free volume from these data. Recalling from Chap. 3 that the loss factor for the mechanical properties occurs at cor = 1, estimate the relaxation time for poly(vinyl acetate) at 40 and 28.5 C. 

Williams and Ferryf measured the dynamic compliance of poly(methyl acrylate) at a number of temperatures. Curves measured at various temperatures were shifted to construct a master curve at 25°C, and the following shift factors were obtained ... 

Master curves can also be constmcted for crystalline polymers, but the shift factor is usually not the same as the one calculated from the WLF equation. An additional vertical shift factor is usually required. This factor is a function of temperature, partly because the modulus changes as the degree of crystaHiuity changes with temperature. Because crystaHiuity is dependent on aging and thermal history, vertical factors and crystalline polymer master curves tend to have poor reproducibiUty. 

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). 

It was shown earlier that the variation of creep or relaxation moduli with time are as illustrated in Fig. 2.9. If we now introduce temperature as a variable then a series of such curves will be obtained as shown in Fig. 2.58. In general the relaxed and unrelaxed modulus terms are independent of temperature. The remainder of the moduli curves are essentially parallel and so this led to the thought that a shift factor, aj, could be applied to move from one curve to another. 

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... 

Solution To use equation (2.76) it would be necessary to know the properties at - 10°C. In this example, the properties are known at 20°C which becomes the reference temperature (J1). The approach taken will be to get the shift factor at T2 — 60°C) and the shift factor at T — 20°C) and then subtract these to get the shift factor from T to 7. 

In order to allow for the effect of temperature on viscosity a shift factor, ar is often used. The Carreau equation then becomes... 

Solution The temperature shift factor from 190°C to 230°C may be obtained from... 

The viscosity flow curves for these materials are shown in Fig. 5.17. To obtain similar data at other temperatures then a shift factor of the type given in equation (5.27) would have to be used. The temperature effect for polypropylene is shown in Fig. 5.2. 

In order to convert data obtained at a given experimental temperature, T, to the reference temperature, Tq, a shift factor, Uj is used, and is defined as... 

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

Investigation of the linear viscoelastic properties of SDIBS with branch MWs exceeding the critical entanglement MW of PIB (about -7000 g/mol ) revealed that both the viscosity and the length of the entanglement plateau scaled with B rather than with the length of the branches, a distinctively different behavior than that of star-branched PIBs. However, the magnitude of the plateau modulus and the temperature dependence of the terminal zone shift factors were found to... 

The glass transition involves additional phenomena which strongly affect the rheology (1) Short-time and long-time relaxation modes were found to shift with different temperature shift factors [93]. (2) The thermally introduced glass transition leads to a non-equilibrium state of the polymer [10]. Because of these, the gelation framework might be too simple to describe the transition behavior. 

Time-temperature superposition [10] increases the accessible frequency window of the linear viscoelastic experiments. It applies to stable material states where the extent of reaction is fixed ( stopped samples ). Winter and Chambon [6] and Izuka et al. [121] showed that the relaxation exponent n is independent of temperature and that the front factor (gel stiffness) shifts with temperature... 

This behavior is in between that of a liquid and a solid. As an example, PDMS properties obey an Arrhenius-type temperature dependence because PDMS is far above its glass transition temperature (about — 125°C). The temperature shift factors are... 

From the PDMS shift factors determined by Winter and Chambon, one may estimate that room temperature fluctuations affect the gel strength by no more than 5%. 

Fig. 9.10 (A) Frequency shift factors aj and (B) modulus shift factor bj as a function of temperature. Reprinted from [40],...

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