Williams, landel and Ferry, WLF equation

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...

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 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) ... 

Since the value of / < 0.159 [120] and T < then it is apparent that in the case of glassy polymers energy of the thermal oscillations of order kT is not sufficient for microvoid formation kT < ej. The second problem requiring explanation and repeatedly discussed [48, 80, 147,148] is the absolute value of f, which within the frameworks of the kinetic theory is estimated according to Equation 1.33. The values 0.050-0.100 were obtained for different polymers [157], which is much more than the generally accepted value of = 0.025 0.003 for most polymers within the frameworks of the Williams, Landel and Ferry concept (WLF) [8, 145,146]. 

This approach is not satisfactory for interpretation of the glass-to-rubber relaxation (a process), as shown in Figure 10. The plot of log/vs. 1/t is distinctly curved, consistent with AH increasing as is approached. This behaviour may be satisfactorily described by an empirical equation proposed by Williams, Landel and Ferry. The WLF equation has been derived from free volume theory and... 

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] ... 

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... 

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... 

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. 

As previously described, the resin shifts the glass transition temperature of the isoprene part. It is possible to explain/predict the shift from the master curves of the pure [SIS-SI] by calculating the modified elementary times Tq and ToHF through Eqs. (16, 17), taking into account the change in the mobility factor (friction coefficient) [5, 25] and considering an increase in the Vogel temperature (Too) of the WLF (Williams, Landel and Ferry [16]) equation. 

A master curve can be constructed as indicated in Figure 22.8, where the zero-shear-rate viscosity t]q has to be evaluated for each one of the indicated viscosity curves. Both, the effect of temperature and pressure on the viscosity versus shear rate curve can be addressed by considering a shift factor that may be related, for instance, to the free volume of the system by means of the Williams, Landel, and Ferry (WLF) equation [9, 15, 23, 24]. With the aid of this shift factor, the new viscosity curve can be constructed from known viscosity values and the reference curve at the prescribed values of temperature and a pressure. The use of shift factors to take into account the temperature dependence on the viscosity curve was also used by Shenoy et al. [19-21] in their methodology for producing viscosity curves from MFI measurements. 

Quite independently, Williams, Landel and Ferry found an empirical equation, now called the WLF equation, which fits the dependence of the shift factor on temperature for a large number of amorphous polymers. The equation is usually written in the form C,(r- Tg)... 

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 ... 

Important here of course is whether the shift factor a-i values calculated from Eq. (24.13) agree with the experimental ones. These results are displayed in Fig. 24.15. The continuous line is calculated from our Eq. (24.13). The dotted line is from an equation proposed in 1955 by Williams, Landel, and Ferry (WLF) [27], a pioneering aj T) formula at that time. We see that the WLF equation works well in a certain temperature range—this seems the reason it is still in use— but fails miserably outside of that range. Nobody else but Ferry [1] stated that range of application of WLF amounts to 50 K or so, not more. If one makes a primitive and unfounded assumption in our Eq. (24.13), one gets from it the WLF equation as a special case [6]. The problem is when people use the WLF equation blindly in wide temperature ranges, obtain bad results, and draw a false conclusion that the time—temperature correspondence principle does not work. 

If the values of shift factor logjo ut, obtained as above, are plotted against test temperature T, a smooth curve is obtained. Williams, Landel and Ferry (1955) showed that the same shift factor-temperature relationship was obtained from the experimental shifting of results from a large number of amorphous polymers. The empirical relationship thus obtained is known as the WLF equation, and can be used in one of the two forms ... 

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