Fulcher equation

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]. 

Several well-known equations are available for interpreting the temperature dependence of viscosity, diffusion coefficient, and other relaxation rates for T > Tg. The Doolittle equation [18], the WLF equation [19], the Vogel-Fulcher equation [20], and the Adam-Gibbs equation [21] can be expressed in the same form. They are known to fit well with the relaxation data of liquids in equilibrium. The universal functional form is [20]... 

The Arrhenius equation did not describe very well the influence of temperature on viscosity data of concentrated apple and grape juices in the range 60-68 °Brix (Rao et al., 1984, 1986). From non-linear regression analysis, it was determined that the empirical Fulcher equation (see Ferry, 1980 p. 289, Soesanto and Williams, 1981) described the viscosity versus temperature data on those juice samples better than the Arrhenius model (Rao et al., 1986) ... 

The magnitudes of the parameters of the Arrhenius and the Fulcher equations for the studied concentrated apple and grape juices are given in Tables 2-6 and 2-7, respectively. The physical interpretation of the three constants in the Fulcher equation is ambiguous, but by translating them in terms of the WLF parameters their significance can be clarified and it is functionally equivalent to the WLF equation (Ferry, 1980 Soesanto and Williams, 1981) ... 

The WLF equation was also used to correlate viscosity versus temperature data on honeys (Al-Malah et al., 2001 Sopade et al., 2003). Because of the empirical nature of the Fulcher equation and the empirical origin of the WLF equation, their use with... 

The temperature dependence of x of a DGEBA oligomer has been investigated. Two types of descriptions for the temperature dependence of X are proposed (Table 6) one is the WLF equation [66], the other is the Vogel-Fulcher equation [64, 65]. The WLF equation for the temperature dependence of X is as follows [66] ... 

As mentioned above, all molecular liquids that have been studied at viscosities above 1 cP (D< 1 X 10 cm sec) have been found to depart the behavior described by (3) in a way that requires the activation energy to increase continuously and in an accelerating manner as the temperature decreases. Over much of the range, the behavior is described by a simple empirical modification of the Arrhenius relation, often called the VTF or Fulcher equation... 

These are the Vogel-Fulcher equations [44]. In addition to the prefactors, two common parameters appear, namely the activation temperature 7, typically 7 = 1000 -2000 K, and the Vogel-Fulcher temperature 7y, whieh is generally 30- 70 K below the glass temperature. Using the Vogel-Fulcher equations, Williams, Landel and Ferry derived an expression for the shift parameter log a. This expression is known in the literature under the name WLF equation [45, 46] ... 

Figure 25 shows the temperature dependence of relaxation time for the relaxation processes in the internal and interfacial regions of the ultrathin PS1.46M film sandwiched between the SiO layers. Since it was hard to distinguish the temperature-Ta relations between the vacuum deposited and laminated films, each data point was averaged over six independent measurements including both vacuum deposited and laminated films. The average thickness was about 40 nm. For comparison, the dashed curve in Fig. 25 denotes the bulk data obtained by the Vogel-Fulcher equation [72, 73] ... 

Fig. 25 Temperature dependence of for aa relaxation processes in internal and interfacial regions of the PS1.46M ultrathin films sandwiched between SiO layers. The average thickness is about 40 nm. The dashed curve denotes the prediction of the Vogel-Fulcher equation using bulk parameters, whereas the solid curve is the best fit by Vogel—Fulcher equation for the interfacial process...

Figure 10. Comparison of the temperature dependence of the chain s diffusion coefficient, D x), and the end-to-end vector scaling time, Tete ( )- Both quantities can be fitted by a Vogel-Fulcher equation (solid lines). Within the error bars the Vogel-Fulcher activation energy and temperature agree with each other for D and Tete- From (53).

Finally, Figure 15 shows the temperature dependence of the inverse relaxation time for the first three Rouse modes. As for the diffusion coefficient and the relaxation time of the end-to-end vector, l/r decreases by about 2 — 3 orders of magnitude in the studied temperature interval and may be fitted by the Vogel-Fulcher equation with a common (but, compared to D and 1/tr, slightly higher)... 

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

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