Thermal activation relations

The temperature-dependence of the viscosity η of a liquid can be described by means of an empirical equation suggested as early as 1921 byVogel: η=A exp.B/(T?T ₀), which, besides the temperatureT, contains three adjustable structuredependent parametersA, B andT ₀. An equation of the same form is suited for describing the temperature dependence of the most probable relaxation timeτ _m in polymers, and can be easily transformed into the socalled WLF-equation, because the empirical parametersB andT ₀ are simple functions of the WLF parameters. In many cases, the temperature-dependence of such quantities as mechanical or dielectric relaxation times, viscosities, diffusion coefficients etc, is regarded as a consequence of a thermal activation process ruled by a temperature-dependentGibbs Free Energy of activationΔG. On the basis of this concept both parametersB andT ₀, can be related to the more fundamental high temperature limiting value of the activation energyΔH and the parameterT ₀ to the temperature at whichΔG becomes infinite. The parameterA is normally taken to be temperature-independent; theoretically, however, it may just as well be temperature-dependent. Although it is difficult to decide from experimental evidence whether this is the case (accurate values over a large temperature range must be available), one should realize that analysis done by means of a temperature-independent pre-exponential factor gives a set of values for the parameters which differ greatly from those found when e.g.A is set proportional toT ^?1 orT ^?2. Literature values for the viscosities ofn-paraffins have been statistically analysed. It is concluded that η in theVogel-equation had better be replaced byη T ^1.2/?, where? is the density of the liquid. Values are given for the modifiedVogel, WLF and activation parameters of the paraffins and ethylenecopolymers, those for the latter being obtained from our own dielectricτ _m measurements. By analogy,τ _m would then have to be replaced byτ _m T ^2.2. However with our dielectric data, such a refinement would not give any great improvement.