Arrhenius equation glass transition

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 WiUiams-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) ... 

A unified approach to the glass transition, viscoelastic response and yield behavior of crosslinking systems is presented by extending our statistical mechanical theory of physical aging. We have (1) explained the transition of a WLF dependence to an Arrhenius temperature dependence of the relaxation time in the vicinity of Tg, (2) derived the empirical Nielson equation for Tg, and (3) determined the Chasset and Thirion exponent (m) as a function of cross-link density instead of as a constant reported by others. In addition, the effect of crosslinks on yield stress is analyzed and compared with other kinetic effects — physical aging and strain rate. 

Viscosity temperature dependence in ILs is more complicated than in most molecular solvents, because most of them do not follow the typical Arrhenius behavior. Most temperature studies fit the viscosity values into the Vogel-Tammarm-Fulcher (VTF) equation, which adds an additional adjustable parameter (glass transition temperature) to the exponential term. 

In this equation C and 02 are primarily empirical constants characteristic of the material and dependent on the chosen reference temperature. They have, however, been given some theoretical interpretation (6). Below the glass transition temperature, Tg, the temperature dependence of the mechanical properties is often described by the Arrhenius equation... 

The variation of tan 8a (referred to the polarizability a) as function of temperature is shown in Fig. 2.79. The activation energy value for this relaxation, calculated according to an Arrhenius equation for the five maxima, is 16kcal mol 1. This is a small value for a glass transition temperature, but not too much considering that in this case a small part of the macromolecule is activated from the dielectric point of view at higher temperatures than that of the p relaxation. 

For amorphous polymers which melt above their glass transition temperature Tg, the WLF equation (according to Williams, Landel, Ferry, Eq. 3.15) with two material-specific parameters q and c2 gives a better description for the shift factors aT than the Arrhenius function according to Eq. 3.14. 

Figure 5 Pure dephasing widths, l/inT]), of the asymmetrical CO-stretching mode of Rh(CO)2acac in DBP versus temperature on a log plot. The solid line through the data is a tit to Equation (4), the sum of a power law and an exponentially activated process. The inset is an Arrhenius plot at higher temperatures showing that the process is activated. Note that there is no break at the experimental glass transition temperature, 169 K. The best fit has the power law exponent, a = 1.0, and the activation energy, AE = 385 cm-1.

The following form of the Arrhenius equation can be used to determine the activation energy for shifting of the glass transition temperature as well as for defining a straight line equation characterizing the shift as a function of frequency. 

Empirical Relationship - Empirical relationships correlating glass transition temperature of an amorphous viscoelastic material with measurement temperature and frequency, such as the William Landel Ferry equation (17) and the form of Arrhenius equation as discussed, assume an affine relationship between stress and strain, at least for small deformations. These relationships cover finite but small strains but do not include zero strain, as is the case for the static methods such as differential scanning calorimetry. However, an infinitely small strain can be assumed in order to extend these relationships to cover the glass transition temperature determined by the static methods (DSC, DTA, dilatometry). Such a correlation which uses a form of the Arrhenius equation was suggested by W. Sichina of DuPont (18). 

Table III also shows that E increases with increasing DSC T. This would be expected from restricted segmental mobility of trie high T samples. Lewis iH found that Arrhenius plots of log frequency versus reciprocal dynamic glass transition temperature for restricted and nonrestricted polymers converges to a different point in the frequency/temperature scale. From this finding, equations were derived to predict static T from the dynamic T value and vice versa. ...

The empirical equation (3), derived from the Arrhenius equation, allows a rough prediction of peak loss factor temperature at dynamic frequencies from DSC glass transition temperature data. 

The Narayanaswamy expression fits volumetric relaxation data well over a range of temperatures for some glass formers (Rekhson et al. 1971 Mazurin 1977 Scherer 1992). But the equation has been criticized for its lack of a physical basis, as well as for its prediction of an Arrhenius temperature-dependence of the relaxation time at equilibrium. Furthermore, near the glass transition, the best-fit value of A // is much larger than the activation energy of the relevant molecular conformational transitions. 

However, there are serious fundamental problems associated with VFT-type equations discussed above, although hardly stated clearly. These equation should be able to transform into Arrhenius-type equations far away from the glass transition. For the VFT eq. (1) r(r) = roexp[Z)7 7 o/(r-ro)] and one obtains r(r) = l instead the Arrhenius equation r(r) = Toexp[ /r] for ro = 0. 

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