Table 3 Hausdorff dimensionality of the bonding system at glass transition.
| Amorphous material | Below Tg (glasses) | Above Tg (supercooled melts) |
|---|---|---|
| Broken bonds – configurons | 0 | 2.5a |
| Chemical bonds backbone cluster | 3 | 3 |
| Chemical bonds | 3 | 2.5a |
a Experimental dimensionality – 2.4–2.8.
Most experimental Tg data have been obtained by differential thermal analysis (DTA), differential scanning calorimetry (DSC), or dilatometry [30], where Tg is generally defined as the temperature at which the tangents to the glass and liquid curves of the relevant property intersect (Chapter 3.2). Heating (cooling) rates for DTA/DSC measurements are typically as high as 10 K/min whereas they are in 3–5 K/min range in dilatometry. As already stated, the glass transition is not abrupt but typically occurs over a few tens of degrees. For not very high cooling rates (q), its dependence on q is given by the Bartenev–Ritland equation:
(11)
where a1 and a2 are empirical 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