Temperature and pressure dependence of ionic conductivity

The dc conductivity, a, of a homogeneous polymer electrolyte, at temperature T, and pressure P, can be expressed in general terms as 

For transport in amorphous systems, the temperature dependence of a number of relaxation and transport processes in the vicinity of the glass transition temperature can be described by the Williams-Landel-Ferry (WLF) equation (Williams, Landel and Ferry, 1955). This relationship was originally derived by fitting observed data for a number of different liquid systems. It expresses a characteristic property, e.g. reciprocal dielectric relaxation time, magnetic resonance relaxation rate, in terms of shift factors, aj, which are the ratios of any mechanical relaxation process at temperature T, to its value at a reference temperature 7, and is defined by 

However, because measurements are kinetically determined, this is a less accurate form of the equation. Very often it is observed that the measured shift factors, defined for different properties, are independent of the measured property. In addition, if for every polymer system, a different reference temperature is chosen, and ap is expressed as a function of T — rj, then ap turns out to be nearly universal for all polymers. Williams, Landel and Ferry believed that the universality of the shift factor was due to a dependence of relaxation rates on free volume. Although the relationship has no free volume basis, the constants and may be given significance in terms of free volume theory (Ratner, 1987). Measurements of shift factors have been carried out on crosslinked polymer electrolyte networks by measuring mechanical loss tangents (Cheradame and Le Nest, 1987). Fig. 6.3 shows values of log ap for 

Eqn (6.5) holds reasonably well for a number of polymer electrolyte systems and a decrease in the leads to an increase in conductivity. 

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 

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