Williams-Landel-Ferry WLF Equation

This equation is the WLF time-temperature superposition principle. When shear compliance versus frequency is plotted for a given frequency, the curves can be superposed at different temperatures by keeping one curve fixed and shifting all other curves along the frequency axis. This superposition creates a shift factor, expressed as [14, 15] 

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Wilkinson hydrogenation, 5 210 Williams-Landel-Ferry (WLF) equation, 21 710... 

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by... 

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]. 

The Arrhenius equation holds for many solutions and for polymer melts well above their glass-transition temperatures. For polymers closer to their T and for concentrated polymer and oligomer solutions, the Williams-Landel-Ferry (WLF) equation (24) works better (25,26). With a proper choice of reference temperature T, the ratio of the viscosity to the viscosity at the reference temperature can be expressed as a single universal equation (eq. 8) ... 

The shift factor is modeled either as a modified Williams-Landel-Ferry (WLF) equation, or as a best fit to the general form of the Equation [20-25]... 

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 -... 

Chow (1980), Condo et al. (1994), Wissinger and Paulaitis (1991), and others. Condo et al. (1994) and Wissinger and Paulaitis (1991) and have shown that the Tg of polystyrene can be reduced to values as low as 35 °C by the addition of about 10 wt % C02. Rudimentary calculations employing the Williams-Landel-Ferry (WLF) equation show that the scaling factors presented in Figure 11.7 are consistent with the Tg measurements cited earlier for the PS-C02 system. 

One of these models was proposed by Wisanrakkit and Gillham (1990). They modified the Williams Landel-Ferry (WLF) equation (Williams et al., 1955), to permit its application both above and below Tg ... 

In some epoxy systems ( 1, ), it has been shown that, as expected, creep and stress relaxation depend on the stoichiometry and degree of cure. The time-temperature superposition principle ( 3) has been applied successfully to creep and relaxation behavior in some epoxies (4-6)as well as to other mechanical properties (5-7). More recently, Kitoh and Suzuki ( ) showed that the Williams-Landel-Ferry (WLF) equation (3 ) was applicable to networks (with equivalence of functional groups) based on nineteen-carbon aliphatic segments between crosslinks but not to tighter networks such as those based on bisphenol-A-type prepolymers cured with m-phenylene diamine. Relaxation in the latter resin followed an Arrhenius-type equation. 

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... 

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... 

T > To are shifted to longer times, and measurements for T < Tq aie shifted to shorter times. A well-defined reduced curve means the viscoelastic response is thermorheologically simple (Schwarzl and Staverman, 1952). It represents log Jp(t) at To over an extended time range. The time scale shift factors aj that were used in the reduction of the creep compliance curves to obtain the reduced curve constitute the temperature dependence, ar is fitted to an analytical form, which is often chosen to be the Williams-Landel-Ferry (WLF) equation (Ferry, 1980),... 

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