Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

measurements made on B₂O₃ (Figure 12a) where three different temperature intervals are distinguished [34]. Above about 270 °C, the liquid phase is in internal thermodynamic equilibrium because its heat capacity is uniquely defined by temperature (and pressure). In the 270–100 °C interval, internal equilibrium is lost as C_p is no longer defined by temperature only. The measurements made upon heating and cooling differ and C_p differences of up to 20% are found between samples initially cooled rapidly and slowly. Also noteworthy is the fact that the observed C_p hysteresis prevents a reversible thermodynamic pathway from being followed. It points instead to the creation of entropy through cycling in this interval and, therefore, demonstrates the irreversibility of the glass–liquid transformation.

Figure 12 Irreversibility of the glass transition: heat capacity hysteresis measured for boron oxide upon cooling and upon heating of a slowly (S) and rapidly (R) cooled glass [34]. (b) Enthalpy and C_p differences between glasses cooled at different rates q; Sup. liq.: enthalpy of the equilibrium supercooled liquid.

Below 100 °C, C_p depends again only on temperature. If integrated from 270 to 100 °C, however, the C_p and C_p /T differences between the rapidly and slowly cooled samples represent enthalpy and entropy differences, respectively. These are constant below 100 °C as the glass C_p does not depend sensitively on thermal history. They can be readily calculated for any two glasses, like a volume difference, if their fictive temperatures are known (Chapter 3.6). An important conclusion then follows: the existence of an entropy difference at 0 K between two samples implies that glasses have a residual entropy at 0 K: hence, glasses do not obey the third Law of thermodynamics because of the irreversible nature of the glass transition (cf. Chapters 3.6 and 10.11).

      In more detail, the C_p 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 C_p variations or even of the endothermic peaks recorded in thermal analysis. This temperature may, for instance, be taken as the inflection point of the C_p 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 T_g and the melting temperature T_f 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 T_g. 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_∞),

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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