The Scaling of the Dynamics of Glasses and Supercooled Liquids
Despite the fact that glasses are materials which have been available since the rise of mankind and despite the fact that they play an essential role in modern technology their physical understanding is still controversial and remains an unresolved proble
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4.1 Introduction Despite the fact that glasses are materials which have been available since the rise of mankind and despite the fact that they play an essential role in modern technology their physical understanding is still controversial and remains an unresolved problem of condensed matter physics [1, 2, 3a–d]. The most prominent features observed when a glass-forming liquid or polymer melt cools down is the rapid increase of the characteristic relaxation time and the strong nonDebye behaviour of the relaxation function. This has been observed by a manifold of different experimental methods including mechanical-dynamical spectroscopy [4], ultrasonic attenuation [5], light [6] and neutron scattering [7] (see Chap. 18), NMR spectroscopy [8] (see Chap. 17) and especially broadband dielectric spectroscopy [9–33]. In the high temperature limit the dielectric relaxation time has a typical value of about τ∞ ≅ 10–13 s, describing local orientational fluctuations. In this range the viscosity of the liquid has a value of about 10–2 to 10–1 poise. With decreasing temperature both the relaxation time and the viscosity increase strongly and can be approximated by the empirical Vogel-Fulcher-Tammann (VFT)-equation [34a–c] 1
ν (T ) =
⎡ − DT0 ⎤ 1 = v∞ exp ⎢ ⎥ 2 π τ (T ) ⎣ T − T0 ⎦
(4.1)
where ν∞ = (2π τ∞)–1, D is a constant and T0 denotes the Vogel temperature. (Sometimes T0 is also called ideal glass transition temperature.) The degree of deviation from an Arrhenius-type temperature dependence provides a useful classification of glass forming systems [35]. Materials are called “fragile” if their ν (T) dependence deviates strongly from an Arrhenius-type behaviour and “strong” if ν (T) is close to the latter. As a quantitative measure of “fragility” the parameter D in Eq. (4.1) can be used [35]. At the calorimetric glass transition Tg ν (Tg) and the viscosity η (Tg) have reached a typical value of ~10–2 Hz and ~1013 poise, respectively. In general, T0 is found to be approximately 40 K below 1
A more refined analysis based on the derivate with respect to temperature (see below) proves that the VFT-equation is a coarse-grained description only.
F. Kremer et al. (eds.) Broadband Dielectric Spectroscopy © Springer-Verlag Berlin Heidelberg 2003
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4 The Scaling of the Dynamics of Glasses and Supercooled Liquids
Tg. Thus, the change in the dynamics of the glass-forming processes spans more than 15 decades. The divergence of Eq. (4.1) at T = T0 is also supported by the so-called Kauzmann paradox occurring in the entropy determined by measurements of the specific heat [36–40]. If the entropy of the supercooled liquid is extrapolated to low temperatures, it seems to become identical to that of a crystal or even smaller at T = T0. In some theories (like the Gibbs-DiMarzio model [41] for polymers) the Kauzmann paradox is resolved by a phase transition. However, the physical meaning of the divergence of ν (T) at T = T0 remains unclear. Because of the universality of Eq. (4.1) T0 should be regarded as a characteristic temperature fo
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