Temperature Vogel

A reference temperature T0 is introduced, which is the equivalent of the Vogel temperature and found empirically in the field of polymer viscosity [60]. Thus in Eq. 131, Tis replaced by (T-T0). Assuming that the limiting shear strain for fracture equals / , the fracture condition for the load rate experiment becomes... 

The strong temperature dependence of x below T serves to define another characteristic temperature of glass formation, the Vogel temperature T o- An astoundingly large class of hquids (ionic, polymeric, biological materials. 

Inserting Eq. (4-16) into Eq. (4-10) gives the Vogel-Fulcher-Tammann-Hesse equation, where now the Vogel temperature To = U/A.2R can be computed from the energy U between trans and gauche states. The values of U obtained indirectly in this way, using... 

Macromolecules 30, 3694 (1997) and filled circles aredata of J. A. Pathake/n/., Macromolecules 32, 2553 (1999). The Vogel temperature is adjusted to make this plot linear and Bjf is determined from the slope. 

Derive the relation between WLF coefficients [B/f and T c in Eq. (8.134)] for two choices of the reference temperature Tq. Hint The Vogel temperature Tcc will not change when the reference temperature is changed, but fo will change. ... 

Polymer melts typically have their Vogel temperature 50 K below their glass transition temperature. 

This result is known as the Vogel equation. (See also Problem 4-6.) For polystyrene, the Vogel temperature T2 is expected to be around 50 °C... 

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. 

Calculations on VFT model show that Vogel temperatures for dry PVA is Tq = 267.2 K then, on the basis of... 

Here, Ty is the so-called Vogel temperature, at which the viscosity diverges to infinity. The parameter Tq is the characteristic time related to molecular vibration and B is the activation temperature. Even in the case of the internal phase, the... 

The constants correspond to coo = C, B = yVflEf, and Tq = T o (7b is the Vogel temperature). With Cohen and Turnbull delivered this free-volume model, a theoretical justification of the empirical VFTH equation and the equivalent Williams-Landel-Ferry (WLF) [Williams et al., 1955] equation as well as of the empirical free-volume models of viscosity [Fox and Flory, 1950 Doolittle, 1951]. 

Shown are AV, defined by Eq. (11.10) m, the minimum number of S-S vacancies in a cluster to show liquidlike behavior, coss, the volume of a vacancy in the S-S lattice yVy> defined in Eq. (11.10) Eq, the temperature where the linearly extrapolated free volume disappears Eq, the Vogel temperature defined in Eq. (11.8) Eg, the glass transition temperature h Tg), the hole free-volume fraction at Tg and the temperature range of the fits. 

It can be observed that the term of (1 -x) describes the contribution of Tf. Here, Tg is the glass transition temperature. denotes the Vogel temperature, defined as (Tg - 50) (°C), tq corresponds to the reference relaxation time at Tg, and B is the loeal slope at Tg of the trace of the time-temperature superposition shift factor in the global WiUiam-Landel-Ferry (WLF) equation [53]. 

With T2 = To and Acp 1/T, the VFT equation is obtained. At the Vogel temperature the configurational entropy vanishes, z(T) diverges like z(T) (T — To) but no information about the absolute size of a CRR can be obtained. The approach of Adam and Gibbs was extended by Donth (1992,2001) to obtain the size of a CRR. Within a fluctuation model a formula was developed which allows to calculate a correlation length (or volume Vcrr) from the height of the step in Cp and the temperature fluctuation 8T of a CRR at Tg as... 

The self-concentration model by Lodge and McLeish can be also employed to describe the relaxation map of a miscible blend system. In reference Mpoukouvalas and Floudas (2008), Floudas et al. report dielectric data for the miscible blend poly (methyl methacrylate) (PMMA) and poly(ethylene oxide) (PEO) (see Fig. 12.22). As expected, with increasing concentration of PEO the relaxation rates shifts to lower temperatures. The data can be modeled by assuming the Vogel temperature To in the VFT equation (see Eq. 12.45) to be dependent on the effective concentration according to Eq. 12.57 ... 

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