Williams-Landel-Ferry equation times

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.)...

An alternative approach to describe nucleation from the amorphous state utilizes the glass transition temperature (Tg) concept (Williams et al. 1955 Slade and Levine 1991). Based on this approach, molecular mobility below Tg is sufficiently limited to kinetically impede nucleation for very long times. Amorphous systems, at temperatures above Tg, nucleate at a rate depending on the temperature difference above Tg. Williams et al. (1955) suggested that the rate of nucleation increases rapidly at temperatures just above Tg according to a kinetic expression given by the WLF (Williams-Landel-Ferry) equation. 

Note that Equation 8 or 9 represents an equivalence between frequency and temperature, which can be expressed as a time-temperature equivalence. The Arrhenius equation is found to be most applicable at lower temperatures. At higher temperatures, a better representation of the equivalence between frequency and temperature is given by the WLF (Williams-Landel-Ferry) equation, which can be written as... 

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... 

Williams-Landel-Ferry equation that relates the value of the shift factor, ax (associated with time-temperature superposition of viscoelastic data), required to bring log-modulus (or log-compliance) vs. time or frequency curves measured at different temperatures onto a master curve at a particular reference temperature. To, usually taken at 50 °C above the glass transition temperature (To = Tg + 50 °C) ... 

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. 

Williams-Landell-Ferry equation (WLF equation). An empirical equation for the time-temperature equivalence of creep and other properties that has been successful with many plastics. It is... 

To calculate the time-temperature superposition shift factor by using the WLF (Williams-Landel-Ferry) equation for polymers at temperatures less than 100°C (232 F) above their Tg [2], use... 

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... 

Physical properties of polymers handbook. Mark JE (ed). Springer, New York, 1996) See Time-Temperature Equivalence and Williams-Landell-Ferry Equation. 

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],... 

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

Williams-Landel-Ferry equation for time-tein)erature supeiposition of mechanical properties... 

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]. 

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 -... 

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... 

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. 

The literature offers empirical expressions that relate free volume to relaxation times. In particular, we refer to the Vogel and Williams-Landel-Ferry (WLF) relations derived from fluidity measurements. These macroscopically defined equations provide relaxation rates (i.e., reciprocal relaxation times, r) as functions of temperature. We can convert these to functions of free volume, /, or lattice-hole fraction, h. Due to the essentially linear dependence of h on T, the mathematical form of the original equation is preserved, and thus one has [Robertson, 1992]... 

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). 

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] ... 

Here tj and t2 represent different time instants, and olj follows the Williams-Landel-Ferry (WLF) empirical equation for polymers. 

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 time-temperature shift factor aj (T) has been successfully fitted by the WLF (Williams-Landel-Ferry) empirical equation (Williams et al. 1955 Ferry 1980) ... 

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