Williams-Landel-Ferry constant

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... 

Yildiz, M.E. and Kokini, J.L. Determination of Williams-Landel-Ferry constants for a food polymer system effect of water activity and moisture content, /. RheoL, 45, 903, 2001. 

In spite of the often large contribution of secondary filler aggregation effects, measurements of the time-temperature dependence of the linear viscoelastic functions of carbon filled rubbers can be treated by conventional methods applying to unfilled amorphous polymers. Thus time or frequency vs. temperature reductions based on the Williams-Landel-Ferry (WLF) equation (162) are generally successful, although usually some additional scatter in the data is observed with filled rubbers. The constants C and C2 in the WLF equation... 

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 constants correspond to coo = C, B = yVflEf, and Tq = T o (7b is the Vogel temperature). With Cohen and Turnbull delivered this free-volume model, a theoretical justification of the empirical VFTH equation and the equivalent Williams-Landel-Ferry (WLF) [Williams et al., 1955] equation as well as of the empirical free-volume models of viscosity [Fox and Flory, 1950 Doolittle, 1951]. 

Free-volume theory Molecular motion involves the availability of vacancies. The vacancy volmne is the free volume, Vp, of the liquid, approximately the difference in volume of the liquid, Vl, and crystalline, 14, forms. Vp is a function of temperature. D is a constant close to unity. The Williams-Landel-Ferry (WLF) equation uses a similar approach in which is the fraction of free volume at Tg, about 0.025, and Pl and Pc are the volumetric thermal expansion coefficients of the liquid and solid, respectively. 

In this equation, a is the conductivity, A is a constant proportional to the number of carrier ions, B is a constant, and To is the temperature at which the configurational entropy of the polymer becomes zero and is close to the glass transition temperature (Tg). The VTF equation fits conductivity rather well over a broad temperature range extending from Tg to about Tg +100 K. Equation [3.2] is an adaptation of the William-Landel-Ferry WLF relationship developed to explain the temperature dependence of such polymer properties as viscosity, dielectric relaxation time and magnetic relaxation rate. The fact that this equation can be applied to conductivity implies that, as with these other properties, ionic... 

The temperature dependence of the relaxation time (r) of polymers in the glass transition region cannot be described by the Arrhenius equation as the In r versus 1/T plot is not linear. This means that the motional activation energy is not a constant but a function of temperature. In this situation, the temperature dependence of the relaxation time can be well described by the William-Landel-Ferry (WLF) equation as follows ... 

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... 

Constant in the Fulcher-Fogel-Tamman and Williams-Landel-Ferry equations Debye relaxation time Heat of sublimation... 

The approximate coincidence found in Ref. [33] between the temperatures T, Tf and the characteristic temperatures for glassy polymers, namely, Tq- the constant in the Fulcher-Fogei-Tamman and Williams-Landel-Ferry equations... 

The shear rheology of LCPs can only be accounted properly if a microstmctural parameter is also included in the conventional rheology models. The Williams-Landel-Ferry (WLF) equation due to its empirical nature is found to be suitable for LCPs. The constants present in the WLF equation can be made to fit any kind of... 

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