Williams-Landel-Ferry principle

There are several analytical tools that provide methods of extrapolating test data. One of these tools is the Williams, Landel, Ferry (WLF) transformation.14 This method uses the principle that the work expended in deforming a flexible adhesive is a major component of the overall practical work of adhesion. The materials used as flexible adhesives are usually viscoelastic polymers. As such, the force of separation is highly dependent on their viscoelastic nature and is, therefore, rate- and temperature-dependent. Test data, taken as a function of rate and temperature, can be expressed in the form of master curves obtained by WLF transformation. This offers the possibility of studying adhesive behavior over a sufficient range of temperatures and rates for most practical applications. Fligh rates of strain may be simulated by testing at lower rates of strain and lower temperatures. 

In principle, a simple bench-drawing test may be used to obtain an impression of the stretchability and of the natural draw ratio of a given polymer. However, as the rate of deformation in the bench test is appreciably lower than under technical drawing conditions, testing should be done below the technical drawing temperature. An impression of the order of magnitude of this temperature difference may be obtained by application of the Williams-Landel-Ferry equation (see Chap. 13). The temperature difference may be more than 20 °C. 

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. 

According to Williams-Landel-Ferry (WLF) principle longer time is equivalent to higher temperature. Thus at higher rates the modulus is shifted to the direction of lower temperature and vice versa [210]. 

Figure 17.15 shows master curves of Ef and tan 8 obtained for the bulk SBR sample by conventional dynamic mechanical analysis (DMA), with the frequency range of 0.05 to 50 Hz over a temperature range of -65 to 445 °C. One returns to the TTS principle, which can be expressed by the Williams-Landel-Ferry (WLF) equation [72] to build the master curves. 

There are two superposition principles that are important in the theory of Viscoelasticity. The first of these is the Boltzmann superposition principle, which describes the response of a material to different loading histories (22). The second is the time-temperature superposition principle or WLF (Williams, Landel, and Ferry) equation, which describes the effect of temperature on the time scale of the response. 

Important here of course is whether the shift factor a-i values calculated from Eq. (24.13) agree with the experimental ones. These results are displayed in Fig. 24.15. The continuous line is calculated from our Eq. (24.13). The dotted line is from an equation proposed in 1955 by Williams, Landel, and Ferry (WLF) [27], a pioneering aj T) formula at that time. We see that the WLF equation works well in a certain temperature range—this seems the reason it is still in use— but fails miserably outside of that range. Nobody else but Ferry [1] stated that range of application of WLF amounts to 50 K or so, not more. If one makes a primitive and unfounded assumption in our Eq. (24.13), one gets from it the WLF equation as a special case [6]. The problem is when people use the WLF equation blindly in wide temperature ranges, obtain bad results, and draw a false conclusion that the time—temperature correspondence principle does not work. 

The so-called WLF (Williams, Landel and Ferry) [13] time-temperature superposition principle and the working relation derived from it give a better... 

In fact, the same increase in joint strength that is obtained with a simple viscoelastic adhesive on increasing the rate of debonding, can be achieved by a suitable reduction in test temperature. This is referred to as the principle of rate-temperature equivalence. For amorphous glass-forming liquids above their glass transition temperature Tg, Williams, Landel, and Ferry (WLF) proposed a universal relationship for the ratio of corresponding test rates at temperatures Tand Tgi ... 

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