Equation Vogel-Fulcher-Tamman

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

The temperature dependence of the conductivity is described by the Vogel—Tamman— Fulcher equation. ... 

Type I. The conductivity of these compounds follows a free volume law in all experimental temperature ranges, reflected by a VTF (Vogel-Tamman-Fulcher) equation ... 

The WLF relation was an extension of the Vogel-Tamman-Fulcher (VTF) empirical equation (Vogel, 1921 Tamman and Hesse, 1926 Fulcher, 1925) which was originally formulated to describe the properties of supercooled liquids and, given in its original form, is... 

To see more clearly the temperature effect on ion conduction, the logarithmic molal conductivity was plotted against the inverse of temperature, and the resultant plots showed apparent non-Arrhenius behavior, which can be nicely fitted to the Vogel— Tamman-Fulcher (VTF) equation ... 

Another approach has been developed by Bruno and Della Monica [24-26], This work takes the Vogel-Tamman-Fulcher (VTF) equation, which has been used to rationalize transport properties in molten salts and glassy electrolytes, and modifies it for nonaqueous solutions. The work follows the development of Angell and co-workers [27,28], who carried out a similar development for aqueous solutions. The expression used is... 

The motion of ions (i.e. conductivity) in polymer electrolytes appears to occur by a liquid-like mechanism in which the movement of ions through the polymer matrix is assisted by the large amplitude segmental motion of the polymer backbone. Ionic conductivity primarily occurs in the amorphous regions of the polymer [4,5]. The temperature dependence of the conductivity of polymer electrolytes is best related by the Vogel-Tamman-Fulcher (VTF) equation... 

The importance of polymer segmental motion in ion transport has already been referred to. Although classical Arrhenius theory remains the best approach for describing ion motion in solid electrolytes, in polymer electrolytes the typical curvature of the log a vs. 1/T plot is usually described in terms of Tg-based laws such as the Vogel-Tamman-Fulcher (VTF) [61] and Williams-Landel-Ferry (WLF) [62] equations. These approaches and other more sophisticated descriptions of ion motion in a polymer matrix have been extensively reviewed [6, 8, 63]. 

In parallel with the Williams-Landel-Ferry (WLF) equation, the Vogel-Tamman-Fulcher (VTF) equation is used to express the temperature dependency of conductivity ... 

For the ILs the curvature of the Arrhenius plots are generally observed and the following Vogel-Tamman-Fulcher (VTF) equation (1) have been conveniently used to interpret the results. 

VFTH Vogel, Fulcher, Tamman, Hesse equation... 

Since most of ILs including polymer systems show upper convex curvature in the Arrhenius plot, and not a straight line, the temperature dependence of the ionic conductivity is expressed by Vogel-Fulcher-Tamman (VFT) equation [7] ... 

At the time of development of free volume theory, two important empirical equations of viscosity were known. They are the Doolittle (1951) equation (3.01) and the Vogel, Tamman and Fulcher (VTF) equation (3.02) (Vogel, 1921, Fulcher, 1923, Tammann and Hesse, 1926), which are given below. 

An equation that does fit the data originated from attempts to fit the temperature dependence of the viscosity. The Vogel-Fulcher, or Vogel-Fulcher-Tamman (VFT), equation for x T), viz. 

---