constants. Although Eq. (11) also results from CPT, it should be replaced by a generalized version at high cooling rates [31]. In addition, CPT predicts that the transition takes place not as a sharp discontinuity, but over a finite temperature interval where the properties of the material depend on time as well as on thermal history.
Following the analysis of [8], we may ask why viscous liquids eventually vitrify instead of remaining in the supercooled liquid state when they escape crystallization. One answer to this question is purely kinetic and relies only on increasingly long relaxation times or increasing viscosities on cooling. The glass transition would result only from the limited timescale of feasible measurements so that any glass would eventually relax to the equilibrium state if experiments could last forever. In fact, a simple thermodynamic argument proposed by Kauzmann [32] indicates that this answer is incorrect. The reason originates in the existence of a configurational contribution that causes the heat capacity of a liquid to be generally higher than that of a crystal of the same composition. As a consequence, the entropy of the liquid decreases on cooling faster than that of the crystal (Figure 4).
If the entropy of the supercooled liquid were extrapolated to temperatures much below Tg, it would become lower at a temperature TK than that of the crystal. Because it is unlikely that an amorphous phase could ever have a lower entropy than an isochemical crystal, the conclusion known as Kauzmann's paradox is that an amorphous phase cannot exist below TK. The temperature of such an entropy catastrophe constitutes the lower bound to the metastability limit of the supercooled liquid. As internal equilibrium cannot be reached below TK , the liquid must undergo a phase transition before reaching this temperature. This is, of course, the glass transition, and Kauzmann's paradox suggests that, although it is kinetic in nature, it anticipates a thermodynamic transition. In other words, CPT treats the glass transition as a true phase transformation although as a nonequilibrium one. The liquid transforms in a continuous way into a glass, which behaves mechanically as a crystalline solid when the motions of atoms become very much frustrated below Tg where the extensive clusters of broken bonds of the liquid are no longer present. The degree of frustration then is actually the same as in a 3‐D crystalline material so that the heat capacity does not show the same high rate of change as in the liquid. This feature is clearly seen both in experiments and as an outcome of the CPT concept (Figure 5). Importantly, CPT yields a universal law for susceptibilities such as heat capacity or thermal expansion near Tg [3, 27]:
Figure 4 Entropy of the amorphous and crystalline phases of diopside, CaMgSi2O6.
Source: After [8].
The liquid transforms into a glass below Tg, therefore the entropy of condensed phase (upper curve) does not follow the dashed line which is an extension of liquid entropy curve below Tg.
Figure 5 Comparison between the heat capacities of amorphous o‐terphenol measured and calculated with configuron percolation theory.
Source: After [3].
(12)
A last feature deserving to be mentioned is the “universal” dependence of the light scattering intensity on the time after a temperature jump in the glass transition range of oxide glasses, which is known as the Bokov effect [33]. The intensity displays a maximum whose height and location on the timescale depends on the previous history of the glass. The Bokov effect is associated with nonequilibrium fluctuations produced by coupling between hydrodynamic modes. Detailed investigations in the past decade have demonstrated that similarities observed in the glass transition region of oxides and polymers account for structural transformations related to the formation of spatially extensive structures, which in turn could be related to clustering effects similar to that envisaged by CPT and other similar models. The Bokov effect thus is providing additional arguments to characterize the glass transition as a second order like phase transformation rather than simply as a slowing down of dynamic processes.
7 Perspectives
Understanding vitrification mechanisms is of great importance either practically or theoretically. Although progress made in this respect has been very impressive, many of the questions remain unresolved. Among them, a central one is that of the glass transition itself, which has a pronounced relaxational, kinetic character in spite of its similarity with a second‐order phase transition in the Ehrenfest sense with volume and entropy continuity, but discontinuities of their derivatives that are used in practice to detect Tg. Discussion about the nature of glass continues. After some lull it has gathered new momentum, especially in the second decade of the new century as the microscopic mechanisms generating the glassy state of matter are still debated. Future developments could be based on computer modeling that does also show the appearance of discontinuities in derivative thermodynamic parameters at the glass transition.
Acknowledgements
The author acknowledges help and advice from R. Doremus, V.L. Stolyarova, P. Poluektov, E. Manykin, W.E. Lee, P. James, R.J. Hand, K.P. Travis, G. Moebus, J.M. Parker, A. Varshneya, O.V. Mazurin, M. Liska, J. Marra, C.M. Jantzen, R. Tournier, C.A. Angell, and D.S. Sanditov.
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