The meaning of the "universal" WLF parameters of glass-forming polymer liquids

Although the Williams-Landell-Ferry (WLF) equation for the segmental relaxation time τ(T) of glass-forming materials is one of the most commonly encountered relations in polymer physics, its molecular basis is not well understood. The WLF equation is often claimed to be equivalent to the Vogel-Fulcher-Tammann (VFT) equation, even though the WLF expression for τ(T) contains no explicit dependence on the fragility parameter D of the VFT equation, while the VFT equation lacks any explicit reference to the glass transition temperature Tg, the traditionally chosen reference temperature in the WLF equation. The observed approximate universality of the WLF parameters C1((g)) and C2((g)) implies that τ(T) depends only on T-Tg, a conclusion that seems difficult to reconcile with the VFT equation where the fragility parameter D largely governs the magnitude of τ(T). The current paper addresses these apparent inconsistencies by first evaluating the macroscopic WLF parameters C1((g)) and C2((g)) from the generalized entropy theory of glass-formation and then by determining the dependence of C1((g)) and C2((g)) on the microscopic molecular parameters (including the strength of the cohesive molecular interactions and the degree of chain stiffness) and on the molar mass of the polymer. Attention in these calculations is restricted to the temperature range (Tg < T < Tg + 100 K), where both the WLF and VFT equations apply.