Williams-Landel-Ferry, WLF

The monomeric friction factors and i have a temperature dependence given by the Williams-Landel-Ferry (WLF) relation, equation (27) ... 

Several attempts have been made to superimpose creep and stress-relaxation data obtained at different temperatures on styrcne-butadiene-styrene block polymers. Shen and Kaelble (258) found that Williams-Landel-Ferry (WLF) (27) shift factors held around each of the glass transition temperatures of the polystyrene and the poly butadiene, but at intermediate temperatures a different type of shift factor had to be used to make a master curve. However, on very similar block polymers, Lim et ai. (25 )) found that a WLF shift factor held only below 15°C in the region between the glass transitions, and at higher temperatures an Arrhenius type of shift factor held. The reason for this difference in the shift factors is not known. Master curves have been made from creep and stress-relaxation data on partially miscible graft polymers of poly(ethyl acrylate) and poly(mcthyl methacrylate) (260). WLF shift factors held approximately, but the master curves covered 20 to 25 decades of time rather than the 10 to 15 decades for normal one-phase polymers. 

Wilkinson hydrogenation, 5 210 Williams-Landel-Ferry (WLF) equation, 21 710... 

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) ... 

Wool and O Connor [33] stated that the self-diffusion coefficient should follow a Williams-Landel-Ferry (WLF) temperature dependence providing that the mode of failure remains the same between samples healed at different temperatures between Tg and Tg + 100°C. Using a reference temperature of 196°C (469 K), the WLF relationship for PI700 polysulfone can be written as follows [38] ... 

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]... 

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 -... 

Fig. 2.48 Self-diffusion of nearly symmetric diblock copolymers measured using forced Rayleigh scattering (Dalvi et al. 1993). (a) Diffusivities, D, for the lower molecular weight PS-PVP sample, which is disordered at these temperatures, have been scaled down by a factor of 0.48, assumming Rouse dynamics (b) D for the lower molecular weight symmetric PEP-PEE diblock copolymer have been scaled down by a factor of 0.40, assuming reptation dynamics. The solid line indicates a fit of the standard Williams-Landel-Ferry (WLF) temperature dependence to the data for the lower molecular weight sample. Values of M are in g mol1.

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 ... 

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. 

Fast relaxation processes ( , 0) show a Williams-Landel-Ferry (WLF) type temperature dependence which is typical for the dynamics of polymer chains in the glass transition range. In accordance with NMR results, which are shown in Fig. 9, these relaxations are assigned to motions of chain units inside and outside the adsorption layer (0 and , respectively). The slowest dielectric relaxation (O) shows an Arrhenius-type behavior. It appears that the frequency of this relaxation is close to 1-10 kHz at 240 K, which was also estimated for the adsorption-desorption process by NMR (Fig. 9) [9]. Therefore, the slowest relaxation process is assigned to the dielectric losses from chain motion related to the adsorption-desorption. 

When a is dependent on T — Tq [that is, a Williams-Landel-Ferry (WLF) dependence] or on T (Arrhenius dependence), then is a linear function of time. 

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