Equation, Arrhenius Williams-Landel-Ferry

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 Arrhenius equation holds for many solutions and for polymer melts well above their glass-transition temperatures. For polymers closer to their T and for concentrated polymer and oligomer solutions, the Williams-Landel-Ferry (WLF) equation (24) works better (25,26). With a proper choice of reference temperature T, the ratio of the viscosity to the viscosity at the reference temperature can be expressed as a single universal equation (eq. 8) ... 

An alternative to constructing the Arrhenius plot log(K) against 1/T is to shift the plots of parameter against time along the time axis to construct a master curve. Use can be made of the Williams, Landel, Ferry (WLF) equation -... 

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. 

Note that Equation 8 or 9 represents an equivalence between frequency and temperature, which can be expressed as a time-temperature equivalence. The Arrhenius equation is found to be most applicable at lower temperatures. At higher temperatures, a better representation of the equivalence between frequency and temperature is given by the WLF (Williams-Landel-Ferry) equation, which can be written as... 

Empirical Relationship - Empirical relationships correlating glass transition temperature of an amorphous viscoelastic material with measurement temperature and frequency, such as the William Landel Ferry equation (17) and the form of Arrhenius equation as discussed, assume an affine relationship between stress and strain, at least for small deformations. These relationships cover finite but small strains but do not include zero strain, as is the case for the static methods such as differential scanning calorimetry. However, an infinitely small strain can be assumed in order to extend these relationships to cover the glass transition temperature determined by the static methods (DSC, DTA, dilatometry). Such a correlation which uses a form of the Arrhenius equation was suggested by W. Sichina of DuPont (18). 

In some epoxy systems ( 1, ), it has been shown that, as expected, creep and stress relaxation depend on the stoichiometry and degree of cure. The time-temperature superposition principle ( 3) has been applied successfully to creep and relaxation behavior in some epoxies (4-6)as well as to other mechanical properties (5-7). More recently, Kitoh and Suzuki ( ) showed that the Williams-Landel-Ferry (WLF) equation (3 ) was applicable to networks (with equivalence of functional groups) based on nineteen-carbon aliphatic segments between crosslinks but not to tighter networks such as those based on bisphenol-A-type prepolymers cured with m-phenylene diamine. Relaxation in the latter resin followed an Arrhenius-type equation. 

Further information can be obtained by DMA if analyses ate run at several temperatures. A transition map is conslmcted fiom the temperatures at which tan ( ) has a peak plotted versus the fierjuency. Specifically, the logarithm of the fi equency used in the measurement is plotted against the inverse temperature of the maximum tan ( ) value. If a straight line is obtained the acfivafion energy can be calculated using the Arrhenius relafionsliip. If the jdot produces a curve, the Williams-Landel-Ferry (WLF) equation should be used for further analysis [243]. 

Normally, the viscosity of a liquid decreases with increasing temperature, as seen in Table 1.6 for pure liquid water. For quantitative expression of the temperature effect on the viscosity, several models, such as the Eyring model, the exponential model,Arrhenius model, and Williams—Landel—Ferry model,have been proposed and validated using experimental data. The typical equation relating kinematic viscosity (i/) of the solution to temperature may be expressed as an Arrhenius form ... 

The temperature dependence of the relaxation time (r) of polymers in the glass transition region cannot be described by the Arrhenius equation as the In r versus 1/T plot is not linear. This means that the motional activation energy is not a constant but a function of temperature. In this situation, the temperature dependence of the relaxation time can be well described by the William-Landel-Ferry (WLF) equation as follows ... 

Relaxation mechanisms of dipoles located in dissimilar environments, or originating from complex forms of molecular or ionic motion, usually exhibit curved Arrhenius diagrams. This curvature is usually interpreted in terms of the semiempirical Williams-Landel-Ferry (WLF) equation (Williams et al. 1955)... 

At temperatures T > (melting temperature), the dependence of viscosity on temperature is controlled by the Arrhenius equation. In most materi als, in the temperature range from to (glass transition temperature), the temperature decrease results in an increase of activation energy ( ), which relates to the fact that molecules do not move as individuals, but in a coordinated maimer. At T > Tg, viscosity is satisfactorily described by the so called VTF (Vogel Fulcher Tammany) equation ijj. = A.exp D.Tq/(T Tq) or WLF (Williams—Landel—Ferry) equation Oj. = exp [Cjg.(T—Tg)]/[C2g (T-Tg)], where ijj, = viscosity at temperature T, j. = ratio of viscosities at T and Tg, or the ratio of relaxation times r and tg at temperatures T and Tg and A, D, Tg, Cjg and are constants. Parameters and are considered universal... 

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

The choice of equation for determining the temperature dependence of MFI is mainly governed by whether T < Tg + 100 otT > Tg + 100. At temperatures relatively closer to Tg, free volume and its changes with temperature play a dominant role. Hence, the (Williams-Landel-Ferry) WLF-type Eq. (4.15) could provide better estimates. At temperatures greater than Tg + 100, the temperature dependence of MFI is decisively affected by overcoming of the forces of inter-molecular interactions, in which case the Arrhenius-type Eq. (4.14) would give better predictions. 

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]. 

Arrhenius plots of the ionic conductivity of amorphous polymer electrolytes, such as PPO-based electrolytes, frequently do not lie on a simple straight line, but rather, on a positively curved line (Fig. 3) [11]. Such curves are well represented by a Williams-Landel-Ferry (WLF) type equation [13] ... 

Adhesion is not an intrinsic property of a materials system, but is dependent on many factors. By now it should come as no surprise that the measurement of adhesion is sensitive to both rate and temperature as all of the other mechanical properties have been. In fact, adhesion can often be transformed by the WLF (Williams, Landel, Ferry) equation or Arrhenius transformation in the same maimer as modulus and other properties. Figure 11.7 shows the transformation of isothermal peel data transformed into a master curve along with the polyester adhesive s shear and tensile strength properties. In another study investigating the effect of temperature and surface treatment on the adhesion of carbon fiber/epoxy systan, five epoxy systems were found to fit an overall master curve when corrected for the material T. This result is quite remarkable and is shown in Fignre 11.8. 

In any case, the Arrhenius equation is not particularly useful at temperatures above Tg + 100 K. The overall temperature-dependence of polymer flexibility at temperatures of Tt to T% + 100 K can be expressed by the empirical Williams, Landel, and Ferry (WLF) equation... 

The Arrhenius equation has been employed as a first approximation in an attempt to define the temperature dependence of physical degradation processes. However, the use of the WLF equation (Eq. 3.6), developed by Williams, Landel, and Ferry to describe the temperature dependence of the relaxation mechanisms of amorphous polymers, appears to have merit for physical degradation processes that are governed by viscosity. 

In this paper, we analyze the effect of fluorine substitution in the polymers listed above by dielectric analysis (DEA), dynamic mechanical analysis (DMA) and stress relaxation measurements. The effect of fluorination on the a relaxation was characterized by fitting dielectric data and stress data to the Williams, Landel and Ferry (WLF) equation. Secondary relaxations were characterized by Arrhenius analysis of DEA and DMA data. The "quasi-equilibrium" approach to dielectric strength analysis was used to interpret the effect of fluorination on "complete" dipole... 

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 ... 

The dependence of on stress for dilVerent temperatures between 170 C and 270 C is shown in Fig. 7.13 for PMMA. The shear thinning characteristics of the curves arethe same that is, if the curves are shifted vertically (superposed at constant stress) the curves superpose. Note the large change in viscosity with temperature. As the temperature is lowered towards ( 100 C) the low-stress temperature dependence of increases dramatically this is a commonly observed etfect in all glassy systems and is rationalized by the theory of Williams. Landel, and Ferry (7.N.1). For most liquids of very low relative molecular mas.s, and for polymers at temperatures more than 100 K above T, the Arrhenius equation (4.N.6) is a good fit for the temperature dependence of viscosity. 

The viscosity of a food is extremely high at temperature Tg or Tg (about 10 Pa.s). As the temperature rises, the viscosity decreases, which means that processes leading to a drop in quality will accelerate. In the temperature range of Tg to about (Tg + 100 °C), the change in viscosity does not follow the equation of Arrhenius (cf. 2.5.4.2), but a relationship formulated by Williams, Landel and Ferry (the WLF equation) ... 

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