Probing the Meaning of the Constant B in the Vogel–Fulcher–Tammann Equation Describing Primary Relaxation in Fragile Glass-Forming Liquids: Case Study of Aspirin

Liquids that avoid crystallizing during cooling from the melt become increasingly viscous. Below the glass transition temperature (Tg), the supercooled liquid transforms into an amorphous solid that essentially ceases to flow on practical time scales. While the viscosity of strong glass-formers like silica obeys classical Arrhenius kinetics, fragile glass-forming liquids like some drug molecules exhibit a super-Arrhenius increase in viscosity as a function of the temperature upon cooling to Tg. For fragile liquids, the result is a temperature-dependent activation energy barrier, Ea(T), that can increase several-fold as the temperature is lowered and that exhibits an experimentally established correlation with a growing length scale. While the underlying source of the universal slow-down in the dynamics during glass formation has mystified scientists for decades, this behavior is captured elegantly by the Vogel–Fulcher–Tammann (VFT) equation as it pertains to different types of glasses over various temperature intervals. Unfortunately, the empirical nature of the constant, B, in that equation limits its utility in both practical and theoretical applications. Building on an established framework (Stillinger, F. H. Relaxation and flow mechanics in “fragile” glass-forming liquids. J. Chem. Phys. 1988, 89, 6461−6469), this work posits that the nucleation and growth of two-dimensional (2D) nanosized disks within the liquid, as it vitrifies, are responsible for the primary (α-) relaxation that causes the steep increase in viscosity upon approaching Tg. The value of B for a given material is determined to be reliant on the following three factors: (1) the activation energy barrier of 2D nucleation for the formation of the cooperatively mobile regions within the supercooled liquid (ΔG*), (2) the extrapolated divergence temperature (T0) where the viscosity approaches infinity, and (3) the classical Arrhenius activation energy barrier, Ea0, at T0. Using the drug aspirin as a prototypical fragile glass-forming liquid and making reasonable assumptions (i.e., using simple models and leveraging existing literature) regarding the surface area and interfacial energy of the concertedly moving regions within the liquid, the values 2237 and 2422 K were obtained from two different equations stemming from the proposed methodology, both of which compare favorably to the experimental value of B ≈ 2526 K.