Williams-Landel-Ferry equation dependence

Although a number of equations have been used to describe the temperature dependence of viscosity of pitch systems(15), the Williams, Landel, Ferry equation (WLF) has received relatively little attention. 

An alternative approach to describe nucleation from the amorphous state utilizes the glass transition temperature (Tg) concept (Williams et al. 1955 Slade and Levine 1991). Based on this approach, molecular mobility below Tg is sufficiently limited to kinetically impede nucleation for very long times. Amorphous systems, at temperatures above Tg, nucleate at a rate depending on the temperature difference above Tg. Williams et al. (1955) suggested that the rate of nucleation increases rapidly at temperatures just above Tg according to a kinetic expression given by the WLF (Williams-Landel-Ferry) equation. 

The dependence of the shift factor aj on temperature can often be fit to an empirical expression known as the WtF (Williams-Landel-Ferry) equation (Williams et al. 1955 Ferry 1980) ... 

Another important result deals with the temperature dependence of the correlation times of the elementary motions, which agrees fairly well with the prediction of the William, Landel, Ferry equation, using the phenomenological coefficients obtained from low frequency viscoelastic measurements. Tlf s means that the elementary motions which are observed by FAD and... 

At temperatures T > (melting temperature), the dependence of viscosity on temperature is controlled by the Arrhenius equation. In most materi als, in the temperature range from to (glass transition temperature), the temperature decrease results in an increase of activation energy ( ), which relates to the fact that molecules do not move as individuals, but in a coordinated maimer. At T > Tg, viscosity is satisfactorily described by the so called VTF (Vogel Fulcher Tammany) equation ijj. = A.exp D.Tq/(T Tq) or WLF (Williams—Landel—Ferry) equation Oj. = exp [Cjg.(T—Tg)]/[C2g (T-Tg)], where ijj, = viscosity at temperature T, j. = ratio of viscosities at T and Tg, or the ratio of relaxation times r and tg at temperatures T and Tg and A, D, Tg, Cjg and are constants. Parameters and are considered universal... 

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

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. 

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... 

Viscosity Relations. Several equations have been proposed to describe the dependence of the viscosity of the system on temperature. For polymer systems the Williams Landel-Ferry (WLF) equation is often used. It reads... 

T > To are shifted to longer times, and measurements for T < Tq aie shifted to shorter times. A well-defined reduced curve means the viscoelastic response is thermorheologically simple (Schwarzl and Staverman, 1952). It represents log Jp(t) at To over an extended time range. The time scale shift factors aj that were used in the reduction of the creep compliance curves to obtain the reduced curve constitute the temperature dependence, ar is fitted to an analytical form, which is often chosen to be the Williams-Landel-Ferry (WLF) equation (Ferry, 1980),... 

The literature offers empirical expressions that relate free volume to relaxation times. In particular, we refer to the Vogel and Williams-Landel-Ferry (WLF) relations derived from fluidity measurements. These macroscopically defined equations provide relaxation rates (i.e., reciprocal relaxation times, r) as functions of temperature. We can convert these to functions of free volume, /, or lattice-hole fraction, h. Due to the essentially linear dependence of h on T, the mathematical form of the original equation is preserved, and thus one has [Robertson, 1992]... 

Equation (10-59) or (10-58) is known as the William-Landels-Ferry (WLF) equation. It applies to all relaxation processes, and therefore also for the temperature dependence of the viscosity (see Section 7.6.4). Its validity is limited to a temperature range from Tg to about Tg + 100 K. Outside this temperature range the expansion coefficient ai varies, not linearly, but with the square root of temperature. 

In this equation, a is the conductivity, A is a constant proportional to the number of carrier ions, B is a constant, and To is the temperature at which the configurational entropy of the polymer becomes zero and is close to the glass transition temperature (Tg). The VTF equation fits conductivity rather well over a broad temperature range extending from Tg to about Tg +100 K. Equation [3.2] is an adaptation of the William-Landel-Ferry WLF relationship developed to explain the temperature dependence of such polymer properties as viscosity, dielectric relaxation time and magnetic relaxation rate. The fact that this equation can be applied to conductivity implies that, as with these other properties, ionic... 

In the above description of local motions, characterizes the segmental modes. In order to know whether these segmental motions observed by NMR in bulk at temperatures well above the glass-transition temperature belong to the glass-transition processes, it is of interest to compare the variations of Ti as a function of temperature with the predictions of the Williams-Landel-Ferry (WLF) equation [19]. The WLF equation describes the frequency dependence of the motional processes associated with the glass-transition phenomena. It can be written as [20]... 

The temperature dependence of the relaxation time (r) of polymers in the glass transition region cannot be described by the Arrhenius equation as the In r versus 1/T plot is not linear. This means that the motional activation energy is not a constant but a function of temperature. In this situation, the temperature dependence of the relaxation time can be well described by the William-Landel-Ferry (WLF) equation as follows ... 

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