William, Landel and Ferry equation

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. 

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. 

Figure 11.6 Schematic representation of relationships between water activity, water content, Tg and viscosity for wheat gluten-based films. Calculated values were obtained using the (GAB) equation [176], Couchman and Karasz equation (CK) [171], and Williams Landel and Ferry equation (WLF) [153]. The critical water activity (aw) and Me are indicated when Tg is equal to the ambient temperature...

The Williams, Landel and Ferry Equation, the Free Volume Theory and Other Related Theories... 

This paper describes the effect of velocity and temperature on the friction coefficient of both filled and unfilled rubber vulcanizates sliding on smooth ice. It has been shown that the mechanism of the friction of rubber on ice is the same as that on other smooth surfaces under similar low sliding speed conditions and that the maximum friction coefficients are similar. The Williams Landel and Ferry equation is used to superpose curves of the velocity dependence of the friction coefficient at different temperatures to produce a master curve and therefore to demonstrate the viscoelastic nature of the frictional mechanism. The frictional behaviour depends on the condition of the ice track and a tentative explanation for this observation is suggested. The frictional properties of vulcanizates containing various amounts of a reinforcing carbon black filler have been studied. 

T, at which the measurements were made and the reference temperature, Tq, according to the Williams, Landel and Ferry equation thus... 

The glass transition temperature is a time (frequency) dependent property, and the property changes associated with the Tg (viscosity, modulus, creep) are also time (frequency) dependent. The relationship of property changes with time/temperature changes has been referred to as time-temperature superposition, and the most common equation relating this superposition is the Williams, Landel and Ferry equation ]14]. 

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. 

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] ... 

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 ... 

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... 

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... 

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) ... 

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 ... 

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 ... 

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. 

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] ... 

There is one more term in both numerator and denominator compared with the WLF equation derived by Williams, Landel and Ferry (Aklonis and Macknight, 1983). The coefficients in eq. (25) are related to temperature and have different meanings than ones in WLF equation, in which these coefficients are treated as constants. The Eq. (25) is the shift factor equation of time coordinate and the expression of time-temperature equivalence of rocks. 

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