Why C₁ = 16-17 in the WLF equation is physical - And the fragility of polymers

From the well-recognized equivalence of the Williams-Landel-Ferry (WLF) equation and the Vogel-Tammann-Fulcher (VTF) equation, τ = τ_o exp (B/[T - T_o]), we shall show that the parameter C₁ in the former is just the number of orders of magnitude between the relaxation time at the chosen reference temperature and the pre-exponent of the VTF equation. Thus C^g _l = log(τ_g/τ_o) (a relation which is not found in the present polymer literature), measures the gap between the two characteristic time scales of the polymer liquid, microscopic and α-relaxation, at the glass transition temperature. For systems which obey these two equations over wide temperature ranges, τ_o is consistent with a quasilattice vibration period in accord with theoretical derivations of the VTF equation and also with the microscopic process of mode coupling theory. Thus for such systems, C^g _l is obliged to have the value 16-17 (depending on how T_g is defined), while C^g _l scaled by T_g will reflect the non-Arrhenius character, i.e. fragility, of the system. In fact when C^g _l has the physical value of 16-17, then (1 - C^g ₂/T_g), which varies between 0 and unity, conveniently gives the 'fragility' of the polymer within the 'strong/fragile' classification scheme. This is useful because it permits prediction from the WLF parameters of other properties such as physical ageing behaviour through the now-established correlation of fragility with other canonical characteristics of glassforming behaviour. Where the best fit C^g _l is not 17 ± 2, the corresponding best fit τ_o must be unphysical, and then the range of relaxation times for which the VTF or WLF equations are valid with a single parameter set will be limited, and the predictions of other properties based on that parameter set will be unreliable.