Williams-Landel-Ferry relation

The monomeric friction factors and i have a temperature dependence given by the Williams-Landel-Ferry (WLF) relation, equation (27) ... 

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]. 

Fast relaxation processes ( , 0) show a Williams-Landel-Ferry (WLF) type temperature dependence which is typical for the dynamics of polymer chains in the glass transition range. In accordance with NMR results, which are shown in Fig. 9, these relaxations are assigned to motions of chain units inside and outside the adsorption layer (0 and , respectively). The slowest dielectric relaxation (O) shows an Arrhenius-type behavior. It appears that the frequency of this relaxation is close to 1-10 kHz at 240 K, which was also estimated for the adsorption-desorption process by NMR (Fig. 9) [9]. Therefore, the slowest relaxation process is assigned to the dielectric losses from chain motion related to the adsorption-desorption. 

Estimation of free-volume parameters for solvent and polymeric membranes Six parameters (three for each solvent and three for the polymer) were estimated using the following theories (a) PDMS (K22 - Tg2> and K22/Y were obtained in literature (Hong, 1995) using polymer viscosity and temperature data. This procedure is expressed in terms of the Williams-Landel-Ferry equation (Williams et al., 1955). The polymer s free volume parameter was related to the Williams-Landel-Ferry constants as presented in equation (2). (b) The same approach was used to obtain (K22 - Tg2) and K22/Y for POMS (equation (2)), but zero shear viscosity data prediction was required prior to this step, (c) EB and Water (K21 - Tgj) and K21/Y parameters were calculated for both components using pure component data of viscosity and temperature (Djojoputro and Ismadji, 2005). Hong (1995) presented equation (3) where free volume... 

Viscosity Relations. Several equations have been proposed to describe the dependence of the viscosity of the system on temperature. For polymer systems the Williams Landel-Ferry (WLF) equation is often used. It reads... 

In this model, the rate constant, k, is expressed as a function of the pre-exponential factor, the ideal gas constant, R, temperature, T, and the activation energy, E. However, the Arrhenius temperature model often falls short of explaining the physical behavior of foods, especially of macro-molecular solutions at the temperatures above T. A better description of the physical properties is offered by the Williams-Landel-Ferry (WLF) model, which is an expression relating the change of the property to the T -T difference [37,38]. That is. 

The literature offers empirical expressions that relate free volume to relaxation times. In particular, we refer to the Vogel and Williams-Landel-Ferry (WLF) relations derived from fluidity measurements. These macroscopically defined equations provide relaxation rates (i.e., reciprocal relaxation times, r) as functions of temperature. We can convert these to functions of free volume, /, or lattice-hole fraction, h. Due to the essentially linear dependence of h on T, the mathematical form of the original equation is preserved, and thus one has [Robertson, 1992]... 

This expression is known as the Williams-Landel-Ferry (WLF) relation. 

However, these two effects are likely to be relatively small compared with the third effect, the variation of the segment fnction factor with composition. An appealing (but inadequate) Ansatz relates changes in the local friction factor with blend composition to changes in the blend glass transition temperature. If we use the Williams-Landel-Ferry form for the temperature dependence of the fnction factor... 

The usual expressions for visco-elastically related properties of amorphous polymers (and of the amorphous regions in semi-crystalline polymers) are the essentially similar Vogel-Tamman-Fulcher (VTF) and Williams-Landel-Ferry (WLF) relationships [30, 45 7]. These can be applied to the dependence of conductivity, a, on absolute temperature, T, for polymer electrolytes, whereupon they have the form... 

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 ... 

Williams-Landel-Ferry equation that relates the value of the shift factor, ax (associated with time-temperature superposition of viscoelastic data), required to bring log-modulus (or log-compliance) vs. time or frequency curves measured at different temperatures onto a master curve at a particular reference temperature. To, usually taken at 50 °C above the glass transition temperature (To = Tg + 50 °C) ... 

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 free volume theories state that the glass transition is characterized by an iso-free volume state, i.e. they consider that the glass temperature is the temperature at which the polymers have a certain universal free volume. The starting point of the theory is that the internal mobility of the system expressed as viscosity is related to the fractional free volume. This empirical relationship is referred to as the Doolittle equation. It is a consequence of the universal William-Landel-Ferry (WLF) equation and the Doolittle equation that the glass transition is indeed an iso-free volume state. The WLF equation, expressed in general terms, is ... 

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... 

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 ... 

La relation de Williams, Landel et Ferry (1955), appelde souvent formule WLF , est une variante de l expression (21) qui elimine la... 

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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