Origin of the Vogel–Fulcher–Tammann law in glass-forming materials: the α–β bifurcation

All glass-forming materials, simple liquids and polymers, show the α–β bifurcation; above a cross-over temperature T∗, the glass α transition and the β secondary transition merge together. Below this bifurcation temperature the relaxation times τ and τβ of the cooperative (α) and non-cooperative (β) movements verify, respectively, the Vogel–Fulcher–Tammann (VFT) and Arrhenius laws. This temperature is of the order of 1.3Tg and in crystallizable materials T∗ is found equal to the melting temperature; the frequency of the α and β motions at that temperature is of the order of 107–109 s−1 depending on the nature of the material. One shows that in this domain, Tg<T<Tg+100°C the cooperativity parameter n (Kohlrausch exponent) of the α movements is of the form n=(T−T0)/(T*−T0), where T0 is a temperature below Tg where the relaxation time τ0 and the exponent n extrapolate, respectively, to infinity and to 0. When the characteristic temperatures T∗ and T0 increase linearly with pressure, then n at constant temperature is also a decreasing function of pressure; 1/n can be considered as the number of individual units (of β type) participating to the α motion, therefore the relaxation time τ verifies the power law: τ=τ0(τβ/τ0)1/n between T0 and T∗, τ0 being the phonon frequency and τβ the frequency of the β movements; this equation is not very different from the Ngai relation concerning the relaxation time of complex systems. Combining both relations n∼ and τ∼(τβ)1/n, one finds that the relaxation time is given by the relation: log τ/τ0≈A/T(T−T0), with A=Eβ(T*−T0)/2.3R; Eβ being the activation energy of the β motions. This law, called modified VFT law, fits the experimental results better than the other phenomenological or theoretical models. This law, without adjustable parameter, is compared to the VFT law obtained if one assumes that the cooperativity parameter n varies as −1/T. The relationships between the fragility index, capacity jump and the ng value at Tg are discussed.