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

Charles Angell

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245 Scopus citations


From the well-recognized equivalence of the Williams-Landel-Ferry (WLF) equation and the Vogel-Tammann-Fulcher (VTF) equation, τ = τo exp (B/[T - To]), 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 Cg l = log(τgo) (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, τo 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, Cg l is obliged to have the value 16-17 (depending on how Tg is defined), while Cg l scaled by Tg will reflect the non-Arrhenius character, i.e. fragility, of the system. In fact when Cg l has the physical value of 16-17, then (1 - Cg 2/Tg), 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 Cg l is not 17 ± 2, the corresponding best fit τo 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.

Original languageEnglish (US)
Pages (from-to)6261-6266
Number of pages6
Issue number26
StatePublished - 1997


  • Polymer fragility
  • WLF parameters

ASJC Scopus subject areas

  • Organic Chemistry
  • Polymers and Plastics
  • Materials Chemistry


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