10.11).
In more detail, the Cp hysteresis results from the observed contrast between a smooth decrease upon cooling and sharp increases upon heating followed by overshoots right at the end of the transition (Figure 12a). The former decrease simply points to the progressive loss of atomic mobility with decreasing temperatures. Upon heating, the situation is more complicated because relaxation resumes at the temperature at which it vanished on cooling, but its first effect is to lower the enthalpy of the glass to bring it closer to the equilibrium values of the supercooled liquid (Figure 12b). At higher temperatures, the enthalpy curve of the material has already crossed that of the supercooled liquid when relaxation becomes almost complete at the timescale of the experiment. The heat capacity then increases rapidly (Figure 12b) in a way that depends on thermal history. The rise is highest for samples initially cooled down at the slowest rates, whose enthalpy is initially the lowest, or for samples heated at the highest rates. If the heating and cooling rates are increased, the transition shifts to higher temperatures because the decrease of the experimental timescale must be matched by an analogous decrease of the relaxation time (Figure 12b).
Determination of a glass‐transition temperature is more complicated in calorimetry than in dilatometry because of the complex shapes of the observed Cp variations or even of the endothermic peaks recorded in thermal analysis. This temperature may, for instance, be taken as the inflection point of the Cp increase upon heating, but it can alternatively be defined in different ways so that is generally needed to specify which particular one has been selected [35].
2.3.5 The Case of Plastic Crystals
This description of the glass transition applies to a variety of kinetically controlled processes in crystals. Plastic crystals, characterized by low entropy of fusion and an unusually high plasticity, are good examples of disordered systems with three‐dimensional long‐range order. When the high‐temperature form of cyclohexanol (ChI), for instance, crystallizes at 299 K, the C6H12O molecules order in a face‐centered cubic lattice but their regular shape allows them to maintain orientational mobility by rotations around the lattice points. It is through a transition to the low‐temperature polymorph (ChII), which is stable below 265 K, that this dynamics vanishes and the orientational disorder disappears [36]. With rapid cooling rates, the ChI form can be obtained metastably and kept for long periods of time below 180 K. On further cooling, a transition is eventually observed near 160 K (Figure 13). The orientational disorder of C6H12O molecules is then frozen in within the crystal. In contrast to the ChII form, whose entropy is zero at 0 K, ChI has a residual entropy of 4.7 J/mol K. The similarity with the glass transition phenomenology is such that the name of glassy crystals has been proposed for crystals where rotation of molecular groups is freed above a glass‐like transition temperature and gives rise to relaxation phenomena much more complex than summarized here (Chapter 8.6).
Figure 13 The calorimetric signature of orientational disorder in cyclohexanol plastic crystal. Measurements made upon heating with a gap from slightly above the glass‐transition temperature Tg and the melting temperature Tf because of rapid transformation into the stable, ordered polymorph.
Source: Data from [36].
2.3.6 Maxwell Model
In view of the continuous pathway between the liquid and glass states, glass‐forming liquids cannot be purely Newtonian when they approach the glass transition. In fact, they are viscoelastic, with an elastic component that becomes increasingly important near Tg. More precisely, application of a shear stress first causes an elastic strain, which would be recovered if the stress were released, and then a viscous deformation. The response of a viscous melt subjected to stress thus is made up of an instantaneous, elastic response along with a delayed response. By combining the simplest representations of elasticity and viscous flow, Maxwell model has as a mechanical analogue a spring and a dash pot placed in series [37]. Its important result is that, if stresses are applied at low frequencies, as usually the case in viscometry, then a simple relationship holds between the viscosity, relaxation time, and shear modulus at infinite frequency (G∞),
(7)
The fact that the glass transition is observed at values close to 1012 Pa.s for widely different kinds of liquids thus indicate that G∞ also weakly depends on composition, with a mean value of about 10 GPa, which varies by less than a factor of 10 with either temperature or composition at least for oxide glass‐forming liquids [38, 39]. Compared with the tremendous variations of viscosity with temperature and composition, G∞ thus is almost constant. If the viscosity is known, structural relaxation times can be readily estimated from Eq. (7).
2.4 Configurational Properties
2.4.1 Equivalence of Relaxation Kinetics
It is usually more difficult to account for the kinetics of a reaction than for its thermodynamics. Relaxation in glass‐forming systems does not depart from this rule. Whereas a single‐order parameter such as the fictive temperature may be appropriate for characterizing the volume or enthalpy of a glass, relaxation kinetics requires models much too complex to be discussed here (see Chapter 3.7). One can nonetheless have a first look at the mechanisms involved in relaxation by examining whether their kinetics varies or not with the particular property considered.
As done for viscosity, the kinetics of volume equilibration can, for instance, be measured by isothermal dilatometry experiments. If samples with the same thermal history are studied, comparisons between the relaxation kinetics of different properties can be made in terms of normalized variables
(8)
where Yt, Y∞, and Y0 are the property Y at time t, initial time, and equilibrium, respectively. To within experimental errors, experiments on E glass, for example, show in this way the same kinetics for viscosity and volume (Figure 14).
