Williams-Landel and Ferry

Williams, Landel and Ferry developed an empirical relationship for this type of shift factor. This has the form... 

The value of Uj. itself is obtained by using the so-called WLF equation (7.18), first proposed by Williams, Landel, and Ferry in 1955. 

The glass transition temperature can be chosen as the reference temperature, though this was not recommended by Williams, Landel, and Ferry, who preferred to use a temperature slightly above T. In order to determine relaxation times, and hence a, use can be made of dynamic mechanical, stress relaxation, or viscosity measurements. 

When the test temperature is raised, the rate of Brownian motion increases by a certain factor, denoted Ox. and it would therefore be necessary to raise the frequency of oscillation by the same factor flx to obtain the same physical response, as shown in Figure 1.6. The dependence of Uj upon the temperature difference T—Tg follows a characteristic equation, given by Williams, Landel, and Ferry (WLF) [11] ... 

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. 

The method of relating the horizontal shifts along the log time scale to temperature changes as developed by Williams, Landel, and Ferry from equation (24) is known as the WI.F method. The amount of horizontal slut of (he log time scale is givvn by log a,-. If the glass transition temperature is chosen as the reference temperature, the temperature dependence ni the shift lactoi lor most amorphous polymers is... 

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by... 

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... 

According to the more widely used Williams, Landel, and Ferry (WLF) equations, all linear, amorphous polymers have similar viscoelastic properties at Tg and at specific temperatures above Tg, such as Tg + 25 K, and the constants Ci and C2 related to holes or free volume, the following relationship holds ... 

Williams, Landel, and Ferry equation (WLF) Used for predicting viscoelastic properties at temperatures above Tg when these properties are known for one specific temperature, yield point Point on a stress-strain curve below which there is reversible recovery. 

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... 

Some applications require the material to remain under constant stress for years, yet it is often not reasonable to conduct such extended time measurements. One approach which circumvents this employs time-temperature superposition. Measurements are obtained over a shorter time span at differing temperatures. A master curve of C as a function of a reduced time tl a where a is a shift factor, is generated, and this allows the results to be extended to longer times. The shift factor is obtained by employing the Williams, Landel, and Ferry (WLF) relationship... 

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... 

Although the results in Fig. 1.32 where shifted to a reference temperature of 298 K (25°C), Williams, Landel and Ferry [14] chose Tref = 243K for... 

In a very important paper, Williams, Landel and Ferry (1955) demonstrated that the temperature dependence of viscosities of a number of pure polymers could be represented accurately by a simple expression, now widely known as the WLF equation, derived from the free volume... 

Williams, Landel and Ferry introduce their famous WLF-equation for describing the temperature dependence of relaxation times as a universal function of T and Tg... 

The generalised formula for the shift factor is, according to Williams, Landel and Ferry (1955) ... 

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 equation has the same form as the well-known WLF equation (Williams, Landel and Ferry, 1955) that correlates the mechanical behaviour of all polymers near their Tg provided we set Tg = Tx (Tz measured by the same method for each polymer). From experimental results one finds that... 

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) ... 

If temp, product > Tg Lyo, the WLF (Williams, Landel, and Ferry) law applies [25,34]. For a similar increase in temperature, this latter law reveals a much more important decrease in the values of the physicochemical parameters than does Arrhenius s law [25]. 

Equation (8.38), empirically formulated by Williams, Landel, and Ferry in the 1950s, is known as the WLF equation (15). Examples of the variation of Qqt with temperature are shown in Figure 8.15. The plots of T — Tq)/ (In flor) against T — Tq are straight lines whose slopes and intercepts are — l/Cj and —C2/C1, respectively. Though an analysis of limited data led to the postulation that and C2 were universal constants at Tg, this assumption was not supported when the results obtained for a wide variety of viscoelastic materials were considered. 

The Doolittle equation [Eq. (8.130)] can be combined with the assumed linear temperature dependence of free volume [Eq. (8.131)] to get the WLF equation, so-named for Williams, Landel, and Ferry, who first applied it to polymer melts in 1955 ... 

Historically, temperature dependence of mechanical and electrical relaxation times were first examined by Williams, Landel and Ferry... 

This exemplifies the experimental difficulties inherent in determining the absolute value of Tg, which is considered in more detail when thermosets are discussed. Of particular interest is the value that a relaxation-dependent property may have when a system is in the vicinity of the glass transition. This is given by the empirical Williams, Landel and Ferry (WLF) equation ... 

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