Shift factor Arrhenius equation

In these circumstances a modified approach to the shift factor can be used and the formal WLF equation abandoned. The shift factor concept is still used but for these situations an Arrhenius equation is fitted to the shift factor aT (not to log aT) ... 

Another result of the kinetic analysis of the 3,2- and 1,2,6-hydride shifts which tends to support the claissical view of the ion is the close similarity in the /I-factors in the Arrhenius equation. is 10 for the 3,2-shift and 10 for the 1,2,6-equilibration process. Both factors are roughly normal and might be expected for rearrangements where the rate-determining step is a relatively slow hydride transfer between carbon atoms in the classical ion. 

This transposition for amorphous polymers is accomplished using a shift factor ( x) calculated relative to a reference temperature T, which may be equal to Tg. The relationship of the shift factor to the reference temperature and some other temperature T, which is between Tg and T + 50 K, may be approximated by the Arrhenius-like Equation. 

Work in groups of three. The shift factor, or, in the WLF Equation [Eq. (5.76)], is actually a ratio of stress relaxation times, f , in the polymer at an elevated temperature, T, relative to some reference temperature. To, and can be related via an Arrhenius-type expression to the activation energy for relaxation, Erei as... 

The mechanical properties of Shell Kraton 102 were determined in tensile creep and stress relaxation. Below 15°C the temperature dependence is described by a WLF equation. Here the polystyrene domains act as inert filler. Above 15°C the temperature dependence reflects added contributions from the polystyrene domains. The shift factors, after the WLF contribution, obeyed Arrhenius equations (AHa = 35 and 39 kcal/mole). From plots of the creep data shifted according to the WLF equation, the added compliance could be obtained and its temperature dependence determined independently. It obeyed an Arrhenius equation ( AHa = 37 kcal/mole). Plots of the compliances derived from the relaxation measurements after conversion to creep data gave the same activation energy. Thus, the compliances are additive in determining the mechanical behavior. 

Figure 3.13 shows the shift factors aT determined from time-temperature superposition as a function of temperature for melts of two semi-crystalline thermoplastics as well as the Arrhenius plot. For the two polyethylenes (HDPE, LDPE), the progression of log ax can be described with the Arrhenius equation. The activation energies can be determined from the slope as Ea(LDPE) 60 kj/mol and Ea(HDPE) 28 kj/mol. Along with polyethylenes (HDPE, LDPE, LLDPE), other significant semi-crystalline polymers are polypropylene (PP), polytetrafluoroethylene (PTFE) and polyamide (PA). 

For amorphous polymers which melt above their glass transition temperature Tg, the WLF equation (according to Williams, Landel, Ferry, Eq. 3.15) with two material-specific parameters q and c2 gives a better description for the shift factors aT than the Arrhenius function according to Eq. 3.14. 

Lifetime predictions of polymeric products can be performed in at least two principally different ways. The preferred method is to reveal the underlying chemical and physical changes of the material in the real-life situation. Expected lifetimes are typically 10-100 years, which imply the use of accelerated testing to reveal the kinetics of the deterioration processes. Furthermore, the kinetics has to be expressed in a convenient mathematical language of physical/chemical relevance to permit extrapolation to the real-life conditions. In some instances, even though the basic mechanisms are known, the data available are not sufficient to express the results in equations with reliably determined physical/chemical parameters. In such cases, a semi-empirical approach may be very useful. The other approach, which may be referred to as empirical, uses data obtained by accelerated testing typically at several elevated temperatures and establishes a temperatures trend of the shift factor. The extrapolation to service conditions is based on the actual parameters in the shift function (e.g. the Arrhenius equation) obtained from the accelerated test data. The validity of such extrapolation needs to be checked by independent measurements. One possible method is to test objects that have been in service for many years and to assess their remaining lifetime. 

Cables specimens aged in air at temperatures between 120 and 200°C showed also a temperature shift factor (aj) that followed the Arrhenius equation with activation energy of ca. 100 kJ mof. Calculation of the remaining lifetime of cables that had been exposed to air for 25 years yielded lifetimes at ca. 40°C that were approximately half of the lifetime obtained by extrapolation of the high temperature (120-200°C) data (73). This assessment was based on 6 service cables. Gillen and coworkers (15-20) have reported similar deviations for a... 

These shift factors are reported in Fig. 11 as a function of the inverse of the absolute temperature. The temperature dependence of these quantities it clearly follows an Arrhenius type equation in the form ... 

Instead of following the WLF equation, values of the shift factor follow an Arrhenius-type relationship with temperature, indicating-that the chain segment mobility is restricted even in the rubbery state. 

In conclusion, the temperature dependence of shift factors for the networks studied here do not follow the WLF equation, but rather an Arrhenius-type relationship. The apparent activation energies are independent of stoichiometric variation [as they are when is varied by changing prepolymer molecular weight (13)]. ... 

The time-temperature or frequency-temperature superposition scheme discussed in Chapter 4, Section B, is applicable to secondary relaxations as well as to the glass transition, assuming that the observed secondary relaxation peaks are well resolved. When the shift factors are obtained for these secondary relaxations, it is found that their temperature dependencies do not obey the WLF equation but follow an equation of the Arrhenius form, that is ,... 

Adding a catalyst to a system changes the rate of the reaction (Section 16-9), hut this Can you use the Arrhenius equation cannot shift the equilihrium in favor of either products or reactants. Because a catalyst (Section 16-8) to show that lowering affects the activation energy of both forward and reverse reactions equally, it changes hoth activation energy barrier increases rate constants hy the same factor, so their ratio, K, does not change. forward and reverse rates by the same... 

Another popular equation for fitting shift factors is based on the Arrhenius... 

Equation (44) is the well-known WLF equation. Universal values of the various physical parameters in Eq. (44) lead to Cf = 17.44 and C = 51.26 K [10]. These are of the same order of magnitude as Cf and C obtained empirically (34 and 80 K for PMMA, for example), and indeed, time-temperature superposition has been found to work well for a wide range of single-phase polymers, with the proviso that it begins to break down for the relatively fast vibrational modes characteristic of the glassy state [12]. Moreover, although superposition may work for T Tg, at temperatures above about 7 + 50 K the shift factors tend to show an Arrhenius dependence rather than following the WLF equation. 

A shift factor below the Tg can be developed using the Arrhenius activation energy equation. 

The shift factors can be correlated with temperature via the Williams-Landel-Ferry (WLF) or the Arrhenius equations [24,25] ... 

From the temperature dependence of the shift factor, the activation energy (AH) can be deduced from the following Arrhenius equation [26,27] ... 

The Arrhenius-equation describes the shift factors by a linear equation ... 

The shift factor aj, shown in Figure 13.2, is found to obey the Arrhenius equation... 

All of these shift factors are quantitatively in good agreement with two Arrhenius equations with different activation energies AH ... 

In this chapter, we have presented the rheological behavior of homopolymers, placing emphasis on the relationships between the molecular parameters and rheological behavior. We have presented a temperature-independent correlation for steady-state shear viscosity, namely, plots of log ri T, Y) r](jiT) versus log or log j.y, where Tq is a temperature-dependent empirical constant appearing in the Cross equation and a-Y is a shift factor that can be determined from the Arrhenius relation for crystalline polymers in the molten state or from the WLF relation for glassy polymers at temperatures between and + 100 °C. 

Horizontally shifting the curves around a reference temperature, i.e. 90 °C, produced the master curve. The x-axis shift factor, aj, for each temperature was plotted as a function of the temperature. The data can be fit using the Arrhenius equation (Equation 8.5) as a mathematical model (as shown in Fig. 8.8) ... 

Figure 7.18 Horizontal shift factors for the UHIVIWPE at various temperatures compared against Arrhenius equation curve fitting. From [12].

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