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WagnerJVGR05.pdf

WagnerJVGR05.pdf

free enthalpy G log(τ

free enthalpy G log(τ α [s]) log(τ Κ [s]) 4 2 0 -2 -4 -6 -8 -10 -12 4 2 0 -2 -4 -6 -8 -10 -12 glass crystal liquid exp Tg Tm TB τ α τ Κ τ α τ Κ T n gas temperature (a) (b) (c) τ Κ,n Fig. 2. Schematic representation to the construction of the TTT-curve (time-temperature-transformation). (a) isobar dependence of the free enthalpy G on temperature T of a gas, a liquid (stable and supercooled liquid), and a crystal as well as the corresponding glass of a pure substance with the boiling point TB, melting point Tm and the glass transition temperature Tg. (b) The hypothetical intersection point between τK and τα is not reached, since (c) τK goes up strongly due to the drastic viscosity increase with τα (after Debenedetti (1996)). (1995), Bartenev and Lomovskoi (1996)). The most common entrances to the modelling of the temperature dependence of the activation energies are: (1) the empirical VFT-equation (Vogel (1921), Fulcher (1925), Tammann and Hesse (1926)) with the adjustable parameters BV FT and T0 (T0 is usually equated with the Kauzmann-temperature (Hodge (1994), Hodge (1997))) Ea(T) = BV FT 1 − T0 . (3) T (2) the Adam-Gibbs-equation (Adam and Gibbs (1965)) with the adjustable parameter BAG and a temperature dependend configurational entropy Sc(T) Ea(T) = kBBAG . (4) Sc(T) 6

(3) the Avramov-equation (Avramov and Milchev (1988)) with a dimensionless activation energy εA, the Avramov fragility index αA and the glass transition temperature Tg at a viscosity of log(η[Pas]) = 12.3 Ea(T) = αA kBεATg T αA−1 . (5) (4) the Williams-Landel-Ferry-equation (WLF, (Williams et al. (1955)) with the parameters AWLF, BWLF and the fragility index m = dln(τα) � � �Tg at d(Tg/T) the glass transition temperature Tg (Boehmer and Angell (1992), Donth (2001)) Ea(T) = mT − AWLFT(T − Tg) . (6) BWLF + (T − Tg) log(Ω) and the Kauzmann-Temperature T0 ≈ (48...815)K are the asymptotes concerning the frequency or temperature (Donth (1981), Donth (1992), Donth (2001), Hodge (1994), Angell et al. (2000)). The mechanical glass transition for small deformations 10 −4 is the linear viscoelastic response of the material. Then the linear response is completely determined by properties of the equilibrium melt if thermal fluctuations are included in the equilibrium concept. The fluctuation-dissipation theorem (FDT) of statistical physics implies the important statement that it is exclusively thermal fluctuations which determine the linear response (Donth (2001), Roling (2001)). Especially for the understanding of processes 1 on a geological time scale these concepts are very useful (Nemilov and Johari (2003)). Nonlinearity has to be take into account for large deformations and high strain rates, i.e. under emplacement conditions (Bagdassarov and Dingwell (1993), Dragoni and Tallarico (1994), Dragoni and Tallarico (1996), Renner et al. (2000), Blake and Bruno (2000), Buisson and Merle (2002)). Above a critical temperature TC predicted by the idealized mode coopling theory, MCT, the system is retained to ergoic and below non-ergodic (Goetze and Sjoegren (1992)). Non-ergodicity means that the correlation function Φ does not converge against an expected equilibrium value of limt→∞ Φ(t) = Φ∞ = 0, but reaches a final value Φ∞ > 0. Since the corresponding coupling parameters are temperature dependent, one finds the critical temperature TC if the existence of a non-ergodic stage to the first time have been observed. In this sense one speaks of a phase transition at the glass transition but The precise designation is egodicity transition. That is, that density fluctuations froze below TC. Above TC the structural a-relaxation and second β-relaxation can not be distinguished. Below TC diffusion processes 1 Within glass science these processes called ’Thermometereffekt’ according to internal friction investigations of Roetger (1941). 7

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