Williams-Landel-Ferry parameters

An alternative to constructing the Arrhenius plot log(K) against 1/T is to shift the plots of parameter against time along the time axis to construct a master curve. Use can be made of the Williams, Landel, Ferry (WLF) equation -... 

For amorphous polymers which melt above their glass transition temperature Tg, the WLF equation (according to Williams, Landel, Ferry, Eq. 3.15) with two material-specific parameters q and c2 gives a better description for the shift factors aT than the Arrhenius function according to Eq. 3.14. 

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... 

Figure 2 shows how glass transition temperatures (Tg) obtained by dynamic mechanical spectroscopy (DMS), percent crystallinities obtained by wide angle x-ray scattering (WAXS) or differential scanning calorimetry (DSC), experimental diffusion coefficients, and information on tortuosity obtained by studies of morphology, can be useful in applying both the theory of V D and the model of P D. The Williams-Landel-Ferry (WLF) parameters [18] c % and C2 , which can be determined by DMS, are needed as additional input for the theory of V D. Densities and thermal expansion coefficients are needed as additional input for the model of P D. 

Initially, a line was constructed for each graph that passed through the points. For analysis of the data, the form of the curve drawn is not important as the Williams Landel Ferry (WLF) transform uses the data points and the parameters for the Arrhenius plot are extracted from the graphs manually, when the best fit can be estimated by eye. 

The K values shown in Table 14.3 for sample C can be well fitted by the Vogel-Tammann-Fulcher (VTF) equation or the Williams-Landel-Ferry (WLF) equation.Prom the VTF equation with the parameters obtained from the fitting, the K values at 127.5 and 93.7gC are calculated and listed in Table 14.3, with the former also listed in Table 14.1. The result of K (andrs) at 93.7gC is used in sections 14.8 and 14.10.a where the structural relaxation time and the length scale at Tg are defined or studied. 

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 (WLF) parameters measurement for trimethylsiloxy-terminated... 

There are two common ways of representing the temperature dependence of the shift factor. One is as an exponential in reciprocal absolute temperature [exp( /Rr)], as done with the polystyrene data this functionality is usually vahd at temperatures well above the glass transition temperature, which is 104 °C for the polystyrene. The polystyrene data are fit with E/R = 9,300 K a value of order 4,000 K is typical of polyethylene, reflecting the wide variation in this parameter. The other commonly used functionahty is the Williams-Landel-Ferry (WLF) equation, which is written... 

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... 

In Giapters 4 and S, a number of unified curves have been presented for a variety of polymers of different generic types. These curves can be readily used for generating specific matraial parameter versus shear rate curves at any required tenqierature of interest merely fiom the knowled of the MFI at that temperature. As ASTM test conditions [1 ] have to be conferred to during MFI test measurements, the MFI at the ASTM test tmnperature needs to be convoted to the MFI value at the required temperature of interest. This could be done through the use of the modified Anfaenhis-type Eq. (4.14) or the modified (Williams-Landel-Ferry) WLF-type Eq. (4.15) described in Sec. 4.2.4. 

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... 

R.E. Robertson Can I ask you a question about that as well Free volume has no totally agreed upon significance and since I ve been working with Simha, I m persuaded that his free volume is a better parameter for some things than the Williams-Landel-Ferry free volume. When you say it s iso free volume, is this iso free volume in the same system, the same free volume definition as Simha uses ... 

If temp, product > Tg Lyo, the WLF (Williams, Landel, and Ferry) law applies [25,34]. For a similar increase in temperature, this latter law reveals a much more important decrease in the values of the physicochemical parameters than does Arrhenius s law [25]. 

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] ... 

Williams, Landel, and Ferry described an empirical equation to represent the shift of time and temperature of the viscoelastic parameters, so that a very wide frequency range can be estimated from a narrow range of measurements, limited by the resonance of the apparatus at the high end and by one s patience and sample degradation change at the low-frequency end [26]. 

To is a reference temperature, which is commonly chosen as the glass transition temperature of the solid, C and B are constants. Williams, Landel and Ferry found that the parameters C = 17.4 and B = 51.6 K were suitable for most polymers they investigated. Changes in viscosity due to an increase in temperature and/or moisture above Tg can be estimated using the WLF equation, if the glass transition temperature is known. 

Williams, Landel and Ferry (1955) have given an empirical expression for a T)y applicable to a wide variety of polymeric materials, for temperatures not too different from the glass transition temperature Tg of the polymer. Their expression contains material-dependent parameters. In its simplest form, there is only one such parameter, the glass transition temperature Tg. This approximate universal expression is given by... 

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