The Williams—Landel—Ferry equation

Long before scientists had developed a fuU treatment of the molecular processes underlying the glass to rubber transition, engineers and technologists needed a formula to convert stress/strain/modulus data measured at one temperature to data at other temperatures.What wanted ras not an involved equation with many com- 

Williams, Landel and Ferry noted that, since time and temperature had an equivalent effect, adjusting the time could accommodate the result of changing the temperature. In other words, for a modulus which was a function of temperature Tj and time f, a new modulus for temperature could be given by  

If this is repeated at different temperatures Tj T2 Tj, etc., and we now move to logarithm (time = reciprocal rate) to convert multiplication by aj to 

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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. 

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). 

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... 

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. 

Another important result is the similarity of the temperature variation of the correlation time r, associated with conformational jumps, and observed for all the polymers considered except polyisobutylene, to the predictions of the Williams-Landel-Ferry equation for viscoelastic relaxation, which indicates that the segmental motions observed by NMR belong to the glass-transition phenomenon. Moreover, the frequency of these intramolecular motions is mainly controlled by the monomeric friction coefficient of the polymer matrix. 

Free Volume and the Williams-Landel-Ferry Equation ... 

A number of experimental methods exist that allow polymer solutions to be subjected to different shear rates or to oscillatory shear. Data obtained over a given range of shear rate, or frequency, are shifted to form a universal curve (as in the use of the Williams-Landel-Ferry equation, explained in Chapter 4). This can then be compared with the predictions of various models such as those proposed by Rouse or Zimm. The former assumes that there is minimal interaction between the solvent and the polymer, and is sometimes referred to as the free draining model. In reality, there is some interaction between the solvent and the polymer chain. This is addressed in the Zimm model, where the drag introduced by the solvent influences the motion of the chains. 

Aii standardized processes, such as lEC 60216 [101] and ISO 2578 [100] (see Chapter 2), consider temperature influence. In both cases, focus on finding the maximum service temperatures rather than extrapolating to normal ambient temperature. The new edition of ISO 11346 [102] for elastomers and thermoplastic elastomers also includes the Williams-Landel-Ferry equation model (Eq. 1.39) for time-temperature shift [94],... 

Tajima and Crozier (14) used the William-Landel-Ferry equation for the variation of epoxy viscosity during cure reaction. The epoxy resin studied was a special resin for pultrusion processing. 

Hence, the results stated above have shown that fluctuation free volume in epoxy polymers possesses fractal structure. Therefor a microvoid forming it should be simulated by D -dimensional sphere. The size of the microvoids is controlled by the volume which is necessary for accumulation of the thermal fluctuations enei y required for their formation. The absolute values of can serve as characteristic of polymer structure thermodynamic non-equilibrium and for quasi-equilibrium structures the value of coincides with the data obtained according to the William-Landel-Ferry equation. Microvoids of fluctuation free volume form fractal structure, which is a mirror of polymer structure [152-158]. 

O) the dashed curve is a guide to the eye through the blend data, and the solid line is the William-Landel-Ferry equation fit to the PVME homopolymer data. 

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