Origin of the Vogel-Fulcher-Tammann law in glass-forming materials: the alpha-beta bifurcation

All glass-forming materials, simple liquids and polymers, show the alpha-be ta bifurcation; above a cross-over temperature T*, the glass alpha transiti on and the beta secondary transition merge together. Below this bifurcation temperature the relaxation times tau and tau(B) of the cooperative (alpha) and non-cooperative (beta) movements verify, respectively, the Vogel-Fulch er-Tammann (VFT) and Arrhenius laws. This temperature is of the order of 1. 3T(g) and in crystallizable materials T* is found equal to the melting temp erature; the frequency of the alpha and beta motions at that temperature is of the order of 10(7)-10(9) s(-1) depending on the nature of the material. One shows that in this domain, T-g < T < T-g + 100 degrees C the cooperati vity parameter n (Kohlrausch exponent) of the alpha movements is of the for m n = (T - T-0)/(T* - T-0), where T-0 is a temperature below T-g where the relaxation time tau(0) and the exponent n extrapolate, respectively, to inf inity and to 0. When the characteristic temperatures T* and T-0 increase li nearly with pressure, then n at constant temperature is also a decreasing f unction of pressure; 1/n can be considered as the number of individual unit s (of beta type) participating to the alpha motion, therefore the relaxatio n time tau verifies the power law: tau = tau(0)(tau(beta)/tau(0))(1/n) betw een T-0 and T*, tau(0) being the phonon frequency and tau(beta) the frequen cy of the beta movements; this equation is not very different from the Ngai relation concerning the relaxation time of complex systems. Combining both relations n similar to T and tau - (tau(beta))(1/n), one finds that the re laxation time is given by the relation: log tau/tau(0) approximate to A/T(T - T-0), with A = E-beta(T* - T-0)/2.3R; E-beta being the activation energy of the beta motions. This law, called modified VFT law, fits the experimen tal results better than the other phenomenological or theoretical models. T his law, without adjustable parameter, is compared to the VFT law obtained if one assumes that the cooperativity parameter n varies as -1/T. The relat ionships between the fragility index, capacity jump and the n(g) value at T -g are discussed. (C) 2000 Elsevier Science B.V. All rights reserved.