Modified WLF equation

In these circumstances a modified approach to the shift factor can be used and the formal WLF equation abandoned. The shift factor concept is still used but for these situations an Arrhenius equation is fitted to the shift factor aT (not to log aT) ... 

The above equation suggests that the Doolittle equation (13) has to be modified to include the nonequilibrium contribution. When 5 = 0, Equation 4 can be written in the form of the WLF equation which is known to be valid for T > Tg. In order to... 

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]... 

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 ... 

The procedure by which the nomograph is generated is not limited to the WLF equation. Since it is based on the reduced variable concept, any superposition equation that results in the calculation of a temperature shift factor may be used to calculate the needed data to create the master curve and subsequent nomograph. The software can easily be modified to calculate and display a master curve on some other superposition equation. 

Chiou and Letton (1992) examine the chemorheology of a complex industrial resin, Hercules 3501-6, containing a major epoxide (TGDDM), two minor epoxides (alicyclic diepoxy carboxylate and epoxy cresol novolac), a hardener (diaminodiphenyl sulfone, DDS) and a Lewis-base catalyst (boron trifluroride monoethylamine complex). They found that both the conventional WLF model (Equation (5.26)) (Ferry, 1980) and a modified WLF model (Equation (5.27) were appropriate to describe the chemorheology ... 

Hou and Bai (1988, 1989) have investigated the chemorheology of Hercules 3501-6 and found good agreement with a modified WLF chemoviscosity model. The equation is summarized as... 

Shen (1990) reviews the flow modelling of injection moulding of thermosets. The kinetic model was the Kamal model (as shown above in Equation (6.31)) and the chemoviscosity model is given by a modified WLF model for low shear mtes (as shown by Equation (6.35)) and a modified Cross model for high shear (as shown by Equation (6.36)), namely... 

Pressure (P) also has an effect on the viscoelastic response. Again, if aU the molecular mechanism involved in the relaxation process has the same pressure dependencies, one may modify the WLF equation to take into account the pressure effect (Moonan and Tschoegl 1985) ... 

The kinetic theory of polymers and the TTSP are only valid above the glass transition temperature. However, many feel that the procedure, in a modified form, is valid below the glass-transition temperature but exactly how far below is uncertain. The WLF equation, on the other hand is known to be only valid above the Tg because below this temperature the material can no longer be considered a super cooled liquid. In fact. Ferry, (1980) notes that the slope of the shift factor curve should be discontinuous at the Tg for the same reason that the coefficient of thermal expansion suffers a discontinuity at the Tg. 

While examining other thermoplastic-modified epoxy systems (ABS-modified DBEBA/DDS system), Jyotishkumar and Thomas et al. [67] recently found that this type of exponential growth of viscosity has connections with the thermoplastic content By comparing the ABS content from 3.6% to 12.9%, the reference relaxation time ts values obtained from the stimulation of WLF equation were found to increases with the ABS content It is believed that a lower viscosity of the epoxy/ DDS in a small content of the ABS-modified systems would favor the polymer chains flowing. 

Note that for copolymers, when MFI values at different temperatures arc required, the modified WLF-type equation cannot be used because of the existence of two glass-transition temperatures in such systems. 

As previously described, the resin shifts the glass transition temperature of the isoprene part. It is possible to explain/predict the shift from the master curves of the pure [SIS-SI] by calculating the modified elementary times Tq and ToHF through Eqs. (16, 17), taking into account the change in the mobility factor (friction coefficient) [5, 25] and considering an increase in the Vogel temperature (Too) of the WLF (Williams, Landel and Ferry [16]) equation. 

However, the L-H model differs in several important macromoleeular details that distinguish it from small moleeule morphology. Following the ad hoe assumption that the first e qjonential in the erystallization equation (A12) may be approximated by a Vogel or modified Williams-Landel-Ferry (WLF) type of term,, Eq. (A 12) beeomes... 

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