Theory Williams-Landell-Ferry

Fig. 3.4. Analysis of the degradation of KS1/ 4 hydrazide conjugate by the Williams-Landel-Ferry glass transition theory (Fig. 9 from [3.10]).

Theories assuming on the formation of a free volume and mobility of the molecules within the holes (Williams, Landel, Ferry, Tobolsky, Kovacs and others)2. ... 

Schatzki, T. F. Theory of measurement of transition temperatures by dilato-metry. I. Glass transition temperature, Williams-Landel-Ferry Approximations. Techn. Rep. n° 55 — 61, Shell Development Co. Emeryville, California... 

Estimation of free-volume parameters for solvent and polymeric membranes Six parameters (three for each solvent and three for the polymer) were estimated using the following theories (a) PDMS (K22 - Tg2> and K22/Y were obtained in literature (Hong, 1995) using polymer viscosity and temperature data. This procedure is expressed in terms of the Williams-Landel-Ferry equation (Williams et al., 1955). The polymer s free volume parameter was related to the Williams-Landel-Ferry constants as presented in equation (2). (b) The same approach was used to obtain (K22 - Tg2) and K22/Y for POMS (equation (2)), but zero shear viscosity data prediction was required prior to this step, (c) EB and Water (K21 - Tgj) and K21/Y parameters were calculated for both components using pure component data of viscosity and temperature (Djojoputro and Ismadji, 2005). Hong (1995) presented equation (3) where free volume... 

Figure 2 shows how glass transition temperatures (Tg) obtained by dynamic mechanical spectroscopy (DMS), percent crystallinities obtained by wide angle x-ray scattering (WAXS) or differential scanning calorimetry (DSC), experimental diffusion coefficients, and information on tortuosity obtained by studies of morphology, can be useful in applying both the theory of V D and the model of P D. The Williams-Landel-Ferry (WLF) parameters [18] c % and C2 , which can be determined by DMS, are needed as additional input for the theory of V D. Densities and thermal expansion coefficients are needed as additional input for the model of P D. 

Free-volume theory Molecular motion involves the availability of vacancies. The vacancy volmne is the free volume, Vp, of the liquid, approximately the difference in volume of the liquid, Vl, and crystalline, 14, forms. Vp is a function of temperature. D is a constant close to unity. The Williams-Landel-Ferry (WLF) equation uses a similar approach in which is the fraction of free volume at Tg, about 0.025, and Pl and Pc are the volumetric thermal expansion coefficients of the liquid and solid, respectively. 

In agreement with the Williams-Landell-Ferry (WLF) theory, the translational mobility of macromolecular segments fully disappears, due to diminishing free volume which reaches zero at temperature To < Tg. For this theory, the following expression is valid ... 

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 ... 

As shown in Figure 6.20, the plots of conductivity vs. 1/T do not obey the Arrhenius-type dependence. The observed convex dependence is characteristic of nonciystalUne phases showing conductivity according to the foregoing mechanism. The conductivity for this case is usually well fitted by a function derived from the free volume theory, called the Williams-Landel-Ferry s (WLF) relationship. 

The importance of polymer segmental motion in ion transport has already been referred to. Although classical Arrhenius theory remains the best approach for describing ion motion in solid electrolytes, in polymer electrolytes the typical curvature of the log a vs. 1/T plot is usually described in terms of Tg-based laws such as the Vogel-Tamman-Fulcher (VTF) [61] and Williams-Landel-Ferry (WLF) [62] equations. These approaches and other more sophisticated descriptions of ion motion in a polymer matrix have been extensively reviewed [6, 8, 63]. 

The brief discussion above shows that the structure of a polymer electrolyte and the ion conduction mechanism are complex. Furthermore, the polymer is a weak electrolyte, whose ions form ion pairs, triple ions, and multidentate ions after its ionic dissociation. Currently, there are several important models that attempt to describe the ion conduction mechanisms in polymer electrolytes Arrhenius theory, the Vogel-Tammann-Fulcher (VTF) equation, the Williams-Landel-Ferry (WLF) equation, free volume model, dynamic bond percolation model (DBPM), the Meyer-Neldel (MN) law, effective medium theory (EMT), and the Nernst-Einstein equation [1]. 

The fractional free volume f, which is the ratio of the free volume to the overall volume, occupies a central position in tr5nng to understand the molecular origins of the temperature dependence of viscoelastic response. The main assumption of the free-volume theory is that the fractional free volume assumes some universal value at the glass transition temperature. The Williams-Landel-Ferry (WLF) equation for the thermal dependence of the viscosity tj of polymer melts is an outgrowth of the kinetic theories based on the free volume and Eyring rate theory (35). It describes the temperature dependence of relaxation times in polymers and other glass-forming liquids above Tg (33-35). The ratio of a mechanical or dielectric relaxation time, Tm or ra, at a temperature T to its value at an arbitrary reference temperature To can be represented by a simple empirical, nearly universal function. 

Very often relaxation data is interpreted in terms of free volume, for example, using the theories of Bueche [19] and Fujita [20]. The idea that free volume governs molecular mobility gives rise to the two most common forms for the temperature dependence of polymer viscoelasticity. If the free volume goes to zero at absolute zero temperature, the equation of Williams-Landel-Ferry (WLF) can be derived [4,17]... 

Time-temperature superposition (tTs) was carried out for these multi-temperature multi-frequency tests based on Williams-Landel-Ferry (WLF) relationship. It considers the equivalency of time and temperature in the context of free volume theory for an activated flow process in viscoelastic materials such as PET. It has been found the tTs holds for the whole temperature/frequenQr range. The master curve generated from tTs is shown in Figure 13 for the 10 wt. % bamboo-PET composite at the 25.0 C reference temperature. The time-temperature superposition shift factor follows Arrhenius temperature dqrendence according to the expression ... 

There are two superposition principles that are important in the theory of Viscoelasticity. The first of these is the Boltzmann superposition principle, which describes the response of a material to different loading histories (22). The second is the time-temperature superposition principle or WLF (Williams, Landel, and Ferry) equation, which describes the effect of temperature on the time scale of the response. 

However, because measurements are kinetically determined, this is a less accurate form of the equation. Very often it is observed that the measured shift factors, defined for different properties, are independent of the measured property. In addition, if for every polymer system, a different reference temperature is chosen, and ap is expressed as a function of T — rj, then ap turns out to be nearly universal for all polymers. Williams, Landel and Ferry believed that the universality of the shift factor was due to a dependence of relaxation rates on free volume. Although the relationship has no free volume basis, the constants and may be given significance in terms of free volume theory (Ratner, 1987). Measurements of shift factors have been carried out on crosslinked polymer electrolyte networks by measuring mechanical loss tangents (Cheradame and Le Nest, 1987). Fig. 6.3 shows values of log ap for... 

The free-volume concept was applied most widely in the theory of viscoelastic properties of polymers developed by Williams, Landel and Ferry (WLF theory), presented in detail in12. According to WLF theory, the changes in liquid viscosity with frequency and temperature from glass temperature T% to T may be plotted on a single master curve by using the reduction factor... 

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 ... 

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