Williams-Landel-Ferry mechanism

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... 

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. 

Figure 2 shows how glass transition temperatures (Tg) obtained by dynamic mechanical spectroscopy (DMS), percent crystallinities obtained by wide angle x-ray scattering (WAXS) or differential scanning calorimetry (DSC), experimental diffusion coefficients, and information on tortuosity obtained by studies of morphology, can be useful in applying both the theory of V D and the model of P D. The Williams-Landel-Ferry (WLF) parameters [18] c % and C2 , which can be determined by DMS, are needed as additional input for the theory of V D. Densities and thermal expansion coefficients are needed as additional input for the model of P D. 

Relaxation mechanisms of dipoles located in dissimilar environments, or originating from complex forms of molecular or ionic motion, usually exhibit curved Arrhenius diagrams. This curvature is usually interpreted in terms of the semiempirical Williams-Landel-Ferry (WLF) equation (Williams et al. 1955)... 

As shown in Figure 6.20, the plots of conductivity vs. 1/T do not obey the Arrhenius-type dependence. The observed convex dependence is characteristic of nonciystalUne phases showing conductivity according to the foregoing mechanism. The conductivity for this case is usually well fitted by a function derived from the free volume theory, called the Williams-Landel-Ferry s (WLF) relationship. 

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. 

The famous Williams-Landell-Ferry (WLF) equation [13] is useful for describing the temperature dependence of several linear mechanical properties of polymers (see Chapter 16). For the zero-shear viscosity, it may be written as... 

Ferry went to Harvard University in 1937 and worked there in a variety of posts, including as a Junior Fellow, until he joined the University of Wisconsin in 1946. He was promoted to Full Professor in 1947 His extensive measurements of the temperature dependence of the dynamic mechanical properties of polymers led to the concept of reduced variables in rheology. His demonstration that time-temperature superposition applied to many systems is the basis for the rational description of polymer rheology. He measured the dynamic response over a very wide range of frequency. One of the fruits of this work is the Williams-Landel-Ferry (WLF) equation for time-temperature shift factors. 

The conductivity of P (MEO-7)/MSCN hybrid film was measured at different temperatures ranging from 0 to 80 °C, and the Arrhenius plot for each system is shown in Fig. 8 [8]. The all of these could be drawn as curved line rather than linear. In other words, the ionic conduction mechanism in each system is considered to obey the Williams-Landel-Ferry(WLF) behavior, in which ionic movement is influenced by the segmental motion of the polymers. 

The brief discussion above shows that the structure of a polymer electrolyte and the ion conduction mechanism are complex. Furthermore, the polymer is a weak electrolyte, whose ions form ion pairs, triple ions, and multidentate ions after its ionic dissociation. Currently, there are several important models that attempt to describe the ion conduction mechanisms in polymer electrolytes Arrhenius theory, the Vogel-Tammann-Fulcher (VTF) equation, the Williams-Landel-Ferry (WLF) equation, free volume model, dynamic bond percolation model (DBPM), the Meyer-Neldel (MN) law, effective medium theory (EMT), and the Nernst-Einstein equation [1]. 

Two additional aspects enhance the utility of the time-temperature superposition concept. First, the same temperature shift factors apply to a particular polymer regar ess of the nature of the mechanical response that is, the shift factors as determined in stress relaxation are applicable to the prediction of the time-temperature behavior in creep or dynamic testing. Second, if the polymer s glass-transition temperature is chosen as the rrference temperature, the shift factors are given by the Williams-Landel-Ferry (WLF) equation in the range Tg[Pg.339]

The fractional free volume f, which is the ratio of the free volume to the overall volume, occupies a central position in tr5nng to understand the molecular origins of the temperature dependence of viscoelastic response. The main assumption of the free-volume theory is that the fractional free volume assumes some universal value at the glass transition temperature. The Williams-Landel-Ferry (WLF) equation for the thermal dependence of the viscosity tj of polymer melts is an outgrowth of the kinetic theories based on the free volume and Eyring rate theory (35). It describes the temperature dependence of relaxation times in polymers and other glass-forming liquids above Tg (33-35). The ratio of a mechanical or dielectric relaxation time, Tm or ra, at a temperature T to its value at an arbitrary reference temperature To can be represented by a simple empirical, nearly universal function. 

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

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. 

The glass transition temperature can be chosen as the reference temperature, though this was not recommended by Williams, Landel, and Ferry, who preferred to use a temperature slightly above T. In order to determine relaxation times, and hence a, use can be made of dynamic mechanical, stress relaxation, or viscosity measurements. 

However, because measurements are kinetically determined, this is a less accurate form of the equation. Very often it is observed that the measured shift factors, defined for different properties, are independent of the measured property. In addition, if for every polymer system, a different reference temperature is chosen, and ap is expressed as a function of T — rj, then ap turns out to be nearly universal for all polymers. Williams, Landel and Ferry believed that the universality of the shift factor was due to a dependence of relaxation rates on free volume. Although the relationship has no free volume basis, the constants and may be given significance in terms of free volume theory (Ratner, 1987). Measurements of shift factors have been carried out on crosslinked polymer electrolyte networks by measuring mechanical loss tangents (Cheradame and Le Nest, 1987). Fig. 6.3 shows values of log ap for... 

This equation has the same form as the well-known WLF equation (Williams, Landel and Ferry, 1955) that correlates the mechanical behaviour of all polymers near their Tg provided we set Tg = Tx (Tz measured by the same method for each polymer). From experimental results one finds that... 

Historically, temperature dependence of mechanical and electrical relaxation times were first examined by Williams, Landel and Ferry... 

The Arrhenius equation has been employed as a first approximation in an attempt to define the temperature dependence of physical degradation processes. However, the use of the WLF equation (Eq. 3.6), developed by Williams, Landel, and Ferry to describe the temperature dependence of the relaxation mechanisms of amorphous polymers, appears to have merit for physical degradation processes that are governed by viscosity. 

In this paper, we analyze the effect of fluorine substitution in the polymers listed above by dielectric analysis (DEA), dynamic mechanical analysis (DMA) and stress relaxation measurements. The effect of fluorination on the a relaxation was characterized by fitting dielectric data and stress data to the Williams, Landel and Ferry (WLF) equation. Secondary relaxations were characterized by Arrhenius analysis of DEA and DMA data. The "quasi-equilibrium" approach to dielectric strength analysis was used to interpret the effect of fluorination on "complete" dipole... 

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