Williams-Landel-Ferry equation temperature effects

Adhesion is not an intrinsic property of a materials system, but is dependent on many factors. By now it should come as no surprise that the measurement of adhesion is sensitive to both rate and temperature as all of the other mechanical properties have been. In fact, adhesion can often be transformed by the WLF (Williams, Landel, Ferry) equation or Arrhenius transformation in the same maimer as modulus and other properties. Figure 11.7 shows the transformation of isothermal peel data transformed into a master curve along with the polyester adhesive s shear and tensile strength properties. In another study investigating the effect of temperature and surface treatment on the adhesion of carbon fiber/epoxy systan, five epoxy systems were found to fit an overall master curve when corrected for the material T. This result is quite remarkable and is shown in Fignre 11.8. 

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... 

Normally, the viscosity of a liquid decreases with increasing temperature, as seen in Table 1.6 for pure liquid water. For quantitative expression of the temperature effect on the viscosity, several models, such as the Eyring model, the exponential model,Arrhenius model, and Williams—Landel—Ferry model,have been proposed and validated using experimental data. The typical equation relating kinematic viscosity (i/) of the solution to temperature may be expressed as an Arrhenius form ... 

When the relaxation frequencies were plotted as a function of the reciprocal temperature, a nonlinear WLF curve of the a-relaxation was obtained for polymers SOP, SOP, and 65P (Figure 2a). Moreover, the relaxation frequency was increased with increasing mesogenic group content, due to the increase of the plasticizer effect. The temperature dependence of the relaxation times has been described by the empirical Williams-Landel-Ferry (WLF) equation (7-8,16)... 

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. 

A master curve can be constructed as indicated in Figure 22.8, where the zero-shear-rate viscosity t]q has to be evaluated for each one of the indicated viscosity curves. Both, the effect of temperature and pressure on the viscosity versus shear rate curve can be addressed by considering a shift factor that may be related, for instance, to the free volume of the system by means of the Williams, Landel, and Ferry (WLF) equation [9, 15, 23, 24]. With the aid of this shift factor, the new viscosity curve can be constructed from known viscosity values and the reference curve at the prescribed values of temperature and a pressure. The use of shift factors to take into account the temperature dependence on the viscosity curve was also used by Shenoy et al. [19-21] in their methodology for producing viscosity curves from MFI measurements. 

Properties also change with temperature, and it is possible to relate time and temperature effects approximately by use of the Williams. Landel, and Ferry (WLF) equation... 

This paper describes the effect of velocity and temperature on the friction coefficient of both filled and unfilled rubber vulcanizates sliding on smooth ice. It has been shown that the mechanism of the friction of rubber on ice is the same as that on other smooth surfaces under similar low sliding speed conditions and that the maximum friction coefficients are similar. The Williams Landel and Ferry equation is used to superpose curves of the velocity dependence of the friction coefficient at different temperatures to produce a master curve and therefore to demonstrate the viscoelastic nature of the frictional mechanism. The frictional behaviour depends on the condition of the ice track and a tentative explanation for this observation is suggested. The frictional properties of vulcanizates containing various amounts of a reinforcing carbon black filler have been studied. 

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