William—Landel—Ferry equation general

The shift factor is modeled either as a modified Williams-Landel-Ferry (WLF) equation, or as a best fit to the general form of the Equation [20-25]... 

In spite of the often large contribution of secondary filler aggregation effects, measurements of the time-temperature dependence of the linear viscoelastic functions of carbon filled rubbers can be treated by conventional methods applying to unfilled amorphous polymers. Thus time or frequency vs. temperature reductions based on the Williams-Landel-Ferry (WLF) equation (162) are generally successful, although usually some additional scatter in the data is observed with filled rubbers. The constants C and C2 in the WLF equation... 

In general, Lewis and Lewis [262] showed that these pseudo-Newtonian pitches follow the Williams, Landel, Ferry (LDF) equation [263] ... 

The master curve in the form of stiffness versus frequency can be created by fitting the experimentally determined shift factors to a mathematical model. With a multifrequency measurement, frequencies beyond the measurable range of the DMA can be achieved by using the superposition method based on the Williams-Landel-Ferry (WLF) equation [60, 61]. For a temperature range above the T, it is generally... 

The free volume theories state that the glass transition is characterized by an iso-free volume state, i.e. they consider that the glass temperature is the temperature at which the polymers have a certain universal free volume. The starting point of the theory is that the internal mobility of the system expressed as viscosity is related to the fractional free volume. This empirical relationship is referred to as the Doolittle equation. It is a consequence of the universal William-Landel-Ferry (WLF) equation and the Doolittle equation that the glass transition is indeed an iso-free volume state. The WLF equation, expressed in general terms, is ... 

Cohen and Turnbull [87] generalized somewhat the theoretical concepts of the relationship between diffusion and self-diffusion of liquids modelled by assemblies of rigid spheres and obtained on the basis of the theories of Frenkel and Eyring, Fox and Flory [88] and Williams, Landell and Ferry [89] the equation ... 

This bottom equation of Equations 13-98 is called the WLF equation, after Williams, Landel and Ferry, who found that for amorphous polymers the curve describing the temperature dependence of the the shift factor aT has the general form (Equation 13-99) ... 

These are the Vogel-Fulcher equations [44]. In addition to the prefactors, two common parameters appear, namely the activation temperature 7, typically 7 = 1000 -2000 K, and the Vogel-Fulcher temperature 7y, whieh is generally 30- 70 K below the glass temperature. Using the Vogel-Fulcher equations, Williams, Landel and Ferry derived an expression for the shift parameter log a. This expression is known in the literature under the name WLF equation [45, 46] ... 

In conclusion, the viscosity of polymer melts depends on shear conditions (rates or stresses), on the molecular weights, and on the temperature. While Newtonian liquids obey an Arrhenius type dependence on temperature, on the other hand, polymer melts follow suit only at temperatures that exceed 100 C above the glass transition temperature (Tg). At the intermediate range, a generalized WLF equation (named after its founders Williams, Landel and Ferry) is applicable ... 

Determination of the shift factor by aligning curves taken at multiple temperatures provides shift factor values only for temperatures at which data are taken. Often it is useful to be able to shift the master curve to any arbitrary temperature, which requires knowledge of the shift factor as a continuous function of temperature. This is generally accomplished by fitting the empirically obtained shift factors to an appropriate function. The most common function used for this purpose is the WiUiams-Landel-Ferry (WLF) equation (Williams et al. 1955) ... 

Since the value of / < 0.159 [120] and T < then it is apparent that in the case of glassy polymers energy of the thermal oscillations of order kT is not sufficient for microvoid formation kT < ej. The second problem requiring explanation and repeatedly discussed [48, 80, 147,148] is the absolute value of f, which within the frameworks of the kinetic theory is estimated according to Equation 1.33. The values 0.050-0.100 were obtained for different polymers [157], which is much more than the generally accepted value of = 0.025 0.003 for most polymers within the frameworks of the Williams, Landel and Ferry concept (WLF) [8, 145,146]. 

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