The Williams-Landel-Ferry WLF equation

Following this empirical discovery there was naturally some speculation as to whether the WLF equation has a ftindamental interpretation. To proceed further we must consider the dilatometric glass transition, and its interpretation in terms of free volume. 

It has subsequently been shown that the original WLF equation can be rewritten in terms of this dilatometric transition temperature such that 

Moreover, it is now possible to give a plausible theoretical basis to the WLF 

The WLF equation can now be obtained in a simple manner. The model representations of linear viscoelastic behaviour all show that the relaxation times are given by expressions of the form r = rjlE (see the Maxwell model in Section 4.2.3 above), where t] is the viscosity of a dashpot and E the modulus of a spring. 

If we ignore the changes in the modulus E with temperature compared with changes in the viscosity rj, this suggests that the shift factor ax for changing temperature from Tg to Twill be given by 

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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) ... 

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),... 

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. 

Equation (10-59) or (10-58) is known as the William-Landels-Ferry (WLF) equation. It applies to all relaxation processes, and therefore also for the temperature dependence of the viscosity (see Section 7.6.4). Its validity is limited to a temperature range from Tg to about Tg + 100 K. Outside this temperature range the expansion coefficient ai varies, not linearly, but with the square root of temperature. 

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. 

Fig. 3.4. With a multi-frequency measurement, frequencies beyond the measurable range of the DMA can be achieved by using the superposition method. Employing the Williams-Landel-Ferry (WLF) equation, and with a treatment of the data, designated as the method of reduced variables or time-temperature superposition (TTS) it is possible to overcome the difficulty of extrapolating limited laboratory tests at shorter times to longer-term, more real world conditions. The underlying bases for TTS are that the processes involved in molecular relaxation or rearrangements in viscoelastic materials occur at accelerated rates at higher temperatures and that there is a direct equivalency between time (the frequency of the measurement) and temperature.

The Williams-Landel-Ferry (WLF) equation (Williams, M. L., et a/., 1955) is probably the most powerful single relationship available for the correlation of viscoelastic behavior in amorphous polymers. Analogous relationships may often be used for semicrystalline and filled polymers. Based on the need of sufficient free volume for chain segments to undergo motion, it interrelates properties such as viscosity and modulus with time (or frequency) and temperature (see Tobolsky, 1960). 

In Figure 12.4 the shift factor log Uj has been plotted vs. temperature. There are two horizontal scales, since the first corresponds to 20°C and the second to the glass transition temperature of the PET-rich phase in the PLC, that is Tg = 62 C. Incidentally, the Tg evident in Figure 12.4 agrees well with values obtained by several other techniques [13,39]. The broken line in the figure has been calculated from the Williams-Landel-Ferry (WLF) equation. The large deviation from experimental values (circles) was expected, since Ferry [37] states that their equation works well around Tg + 50 K, while here an attempt was made to use it below Tg. There are also other problems with the WLF equation, as discussed by Brostow in Chapter 10 of reference [38]. 

Further information can be obtained by DMA if analyses ate run at several temperatures. A transition map is conslmcted fiom the temperatures at which tan ( ) has a peak plotted versus the fierjuency. Specifically, the logarithm of the fi equency used in the measurement is plotted against the inverse temperature of the maximum tan ( ) value. If a straight line is obtained the acfivafion energy can be calculated using the Arrhenius relafionsliip. If the jdot produces a curve, the Williams-Landel-Ferry (WLF) equation should be used for further analysis [243]. 

Since overall diffusion is governed by the diffusion of chain segments, the overall diffusion coefficient, D, is expected to be inversely proportional to the relaxation time of polymer segments [108], which enables a model based on the free volume concept and a description similar to the Williams-Landel-Ferry (WLF) equation [109-112] ... 

In the above description of local motions, characterizes the segmental modes. In order to know whether these segmental motions observed by NMR in bulk at temperatures well above the glass-transition temperature belong to the glass-transition processes, it is of interest to compare the variations of Ti as a function of temperature with the predictions of the Williams-Landel-Ferry (WLF) equation [19]. The WLF equation describes the frequency dependence of the motional processes associated with the glass-transition phenomena. It can be written as [20]... 

For amorphous polymers, the Williams-Landel-Ferry (WLF) equation is often used ... 

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