Williams-Landel-Ferry equation, modeling

Aii standardized processes, such as lEC 60216 [101] and ISO 2578 [100] (see Chapter 2), consider temperature influence. In both cases, focus on finding the maximum service temperatures rather than extrapolating to normal ambient temperature. The new edition of ISO 11346 [102] for elastomers and thermoplastic elastomers also includes the Williams-Landel-Ferry equation model (Eq. 1.39) for time-temperature shift [94],... 

Once the shift factor has been determined for a large enough number of curves, the shift factors themselves may be fitted to a model. This allows the determination of shift factors and master curves at arbitrary temperatures. For viscosity curves taken within 100°C of a material s glass transition temperature (Tg), the WLF (Williams-Landel-Ferry) equation [17] is used ... 

A number of experimental methods exist that allow polymer solutions to be subjected to different shear rates or to oscillatory shear. Data obtained over a given range of shear rate, or frequency, are shifted to form a universal curve (as in the use of the Williams-Landel-Ferry equation, explained in Chapter 4). This can then be compared with the predictions of various models such as those proposed by Rouse or Zimm. The former assumes that there is minimal interaction between the solvent and the polymer, and is sometimes referred to as the free draining model. In reality, there is some interaction between the solvent and the polymer chain. This is addressed in the Zimm model, where the drag introduced by the solvent influences the motion of the chains. 

Moreover, real polymers are thought to have five regions that relate the stress relaxation modulus of fluid and solid models to temperature as shown in Fig. 3.13. In a stress relaxation test the polymer is strained instantaneously to a strain e, and the resulting stress is measured as it relaxes with time. Below the a solid model should be used. Above the Tg but near the 7/, a rubbery viscoelastic model should be used, and at high temperatures well above the rubbery plateau a fluid model may be used. These regions of stress relaxation modulus relate to the specific volume as a function of temperature and can be related to the Williams-Landel-Ferry (WLF) equation [10]. 

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]... 

One of these models was proposed by Wisanrakkit and Gillham (1990). They modified the Williams Landel-Ferry (WLF) equation (Williams et al., 1955), to permit its application both above and below Tg ... 

The constants correspond to coo = C, B = yVflEf, and Tq = T o (7b is the Vogel temperature). With Cohen and Turnbull delivered this free-volume model, a theoretical justification of the empirical VFTH equation and the equivalent Williams-Landel-Ferry (WLF) [Williams et al., 1955] equation as well as of the empirical free-volume models of viscosity [Fox and Flory, 1950 Doolittle, 1951]. 

Since overall diffusion is governed by the diffusion of chain segments, the overall diffusion coefficient, D, is expected to be inversely proportional to the relaxation time of polymer segments [108], which enables a model based on the free volume concept and a description similar to the Williams-Landel-Ferry (WLF) equation [109-112] ... 

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 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 shear rheology of LCPs can only be accounted properly if a microstmctural parameter is also included in the conventional rheology models. The Williams-Landel-Ferry (WLF) equation due to its empirical nature is found to be suitable for LCPs. The constants present in the WLF equation can be made to fit any kind of... 

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]. 

However, the L-H model differs in several important macromoleeular details that distinguish it from small moleeule morphology. Following the ad hoe assumption that the first e qjonential in the erystallization equation (A12) may be approximated by a Vogel or modified Williams-Landel-Ferry (WLF) type of term,, Eq. (A 12) beeomes... 

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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