Generalized Vogel law for glass-forming liquids

Abstract

A model for non-Arrhenius structural and dielectric relaxation in glass-forming materials is based on defect clustering in supercooled liquids. Relaxation in the cold liquid is highly hindered, and assumed to require the presence of a mobile defect to loosen the structure near it. A mild distribution of free-energy barriers impeding defect hopping can generate a wide distribution of waiting times between relaxation events. When the mean waiting time is longer than the time of an experiment, no characteristic time scale exists. This case directly yields the Kohlrausch-Williams-Watts (KWW) relaxation law. A free-energy mismatch between defect and nondefect regions produces a defect-defect attraction, which can lead to aggregation. This may occur in defect-rich “fragile” liquids which also exhibit Vogel kinetics. Defect aggregation and correlation in the “high-temperature” region above the critical consolute temperatureT _c is described using the Ornstein-Zernike theory of critical fluctuations. For a defect correlation length divergence (T-T _c)^-γ/2, a generalized Vogel law for the structural relaxation time τ results: τ=τ₀exp[B./(T-T _c)^1.5γ] In the mean-field limit (γ=1) this provides as good an account of dielectric and structural relaxation in glycerol,n-propanol, andi-butyl bromide as does the original Vogel law, and for the mixed salt KNO₃−Ca(NO₃)₂ and B₂O₂ it also describes kinetics over their entire temperature ranges. A breakdown of the Vogel law in the immediate vicinity ofT _g is avoided, and the need to invoke extra low-temperature mechanisms to explain an apparent “return to Arrhenius behavior” is removed.