William—Landel—Ferry equation

Figure C2.1.15. Schematic representation of tire typicai compiiance of a poiymer as a function of temperature. (C) VOGEL-FULCHER AND WILLIAMS-LANDEL-FERRY EQUATIONS...

The viscosity in the vicinity of Tg follows the Williams-Landel-Ferry equation [3.9], as well as the stability of KS 1/4-DAVBL (deacetylvinblastin hydrazide conjugate), as shown in Fig. 3.4 [3.10]. 

The most popular and widely used formula for the function aT(T) is the Williams-Landel-Ferry equation, which is quite adequate for amorphous polymers above the glass transition temperature ... 

Fig. 25. Shito s test plot of the Williams-Landel-Ferry equation for the dipole relaxation time in an anhydride-cured epoxy. (Reprinted from Ref.50) with permission of John Wiley and Sons, Inc.)...

Bestul, A. B. Application of the Williams-Landel-Ferry equation to silicate glasses. Glastechn. Ber. 32 K, V1/59-VI/66 (1959). 

Although a number of equations have been used to describe the temperature dependence of viscosity of pitch systems(15), the Williams, Landel, Ferry equation (WLF) has received relatively little attention. 

In principle, a simple bench-drawing test may be used to obtain an impression of the stretchability and of the natural draw ratio of a given polymer. However, as the rate of deformation in the bench test is appreciably lower than under technical drawing conditions, testing should be done below the technical drawing temperature. An impression of the order of magnitude of this temperature difference may be obtained by application of the Williams-Landel-Ferry equation (see Chap. 13). The temperature difference may be more than 20 °C. 

An alternative approach to describe nucleation from the amorphous state utilizes the glass transition temperature (Tg) concept (Williams et al. 1955 Slade and Levine 1991). Based on this approach, molecular mobility below Tg is sufficiently limited to kinetically impede nucleation for very long times. Amorphous systems, at temperatures above Tg, nucleate at a rate depending on the temperature difference above Tg. Williams et al. (1955) suggested that the rate of nucleation increases rapidly at temperatures just above Tg according to a kinetic expression given by the WLF (Williams-Landel-Ferry) equation. 

Note that Equation 8 or 9 represents an equivalence between frequency and temperature, which can be expressed as a time-temperature equivalence. The Arrhenius equation is found to be most applicable at lower temperatures. At higher temperatures, a better representation of the equivalence between frequency and temperature is given by the WLF (Williams-Landel-Ferry) equation, which can be written as... 

Empirical Relationship - Empirical relationships correlating glass transition temperature of an amorphous viscoelastic material with measurement temperature and frequency, such as the William Landel Ferry equation (17) and the form of Arrhenius equation as discussed, assume an affine relationship between stress and strain, at least for small deformations. These relationships cover finite but small strains but do not include zero strain, as is the case for the static methods such as differential scanning calorimetry. However, an infinitely small strain can be assumed in order to extend these relationships to cover the glass transition temperature determined by the static methods (DSC, DTA, dilatometry). Such a correlation which uses a form of the Arrhenius equation was suggested by W. Sichina of DuPont (18). 

The dependence of the shift factor aj on temperature can often be fit to an empirical expression known as the WtF (Williams-Landel-Ferry) equation (Williams et al. 1955 Ferry 1980) ... 

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... 

Another important result deals with the temperature dependence of the correlation times of the elementary motions, which agrees fairly well with the prediction of the William, Landel, Ferry equation, using the phenomenological coefficients obtained from low frequency viscoelastic measurements. Tlf s means that the elementary motions which are observed by FAD and... 

C) VOGEL-FULCHER AND WILLIAMS-LANDEL-FERRY EQUATIONS... 

Wentzel-Kramers-Brillouin (approximation) Williams-Landel-Ferry (equation) weight... 

Moreover, T is similar with Tq in the Williams-Landel-Ferry equation or in the Fulcher-Vogel-Tamman equation [14]. The To value is the lowest thermo-... 

The free-volume parameters are again obtained by fitting viscosity versus temperature data using either the adopted Doohttle expression (low-molecular-weight species) or the Williams-Landel-Ferry equation (polymers). The glass transition temperature, Tg, is as reported in the literature or can be estimated from the melting temperature. 

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) ... 

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