near 1100 °C when the ultrasonic velocity becomes independent of frequency.
Figure 9 Frequency dependence of the glass transition range. (a) Compressional acoustic‐wave velocities of sodium disilicate measured at the frequencies (MHz) indicated
(Source: Data from [29]);
larger width of the glass transition range than in dilatometry because of the actual distribution of relaxation times. (b) Compressional hypersonic sound velocities measured for 36 SiO2·16 Al2O3·48 CaO melt (mol %) by Brillouin scattering and ultrasonic methods.
Source: Data from [30, 31].
Experiments can be made at even shorter timescales when hypersonic sound velocities are measured by Brillouin inelastic scattering of photons by phonons (Chapter 2.2). At the timescales of the order of 10−10 seconds of these interactions, the glass transition shifts to higher still temperatures. For calcium aluminosilicates (Figure 9b), relaxed compressional velocities are typically observed only above 2200 °C [30] where they begin to match the values determined by ultrasonic methods (Figure 9b). The first effect noticed when the temperature is increased is a slight kink (at around 750 °C in Figure 9b), which disappears if the velocities are plotted against the volume of the sample instead of its temperature. This kink thus signals the increase in thermal expansion at the volume glass transition, whereas structural relaxation at the extremely short timescale of Brillouin scattering experiments becomes significant only at much higher temperatures. Interestingly, the shear sound velocities can then be measured for the supercooled liquid well above the standard glass‐transition temperature as long as its viscosity is not too low [32]. The material is not really a “glass” because its configuration changes rapidly with temperature, but a “glass‐like” material whose solid‐like part of its acoustic properties may be probed. Finally, another noteworthy feature of the glass transition range is its markedly increasing width apparent from Figures 8b to 9a and b, which originates in the fact that a distribution of relaxation times, and not a single time, must be considered. Complete relaxation is thus controlled by the slowest mechanisms whose retarding effects are the greatest for the shortest experimental timescales.
In conclusion, the question as to whether a given substance is a liquid or a glass cannot be answered if the observational timescale is not specified. One must consider instead that the transition between the two kinds of phases is represented by a curve in the timescale–temperature plane (Figure 10). The picture is actually still more complex because the glass transition also depends on pressure. With the exception of some open 3‐D network structures, Tg generally increases with pressure because an increasing compaction makes configurational rearrangements more difficult. At constant timescale, the glass transition is thus represented by another curve in the pressure–temperature plane (Figure 11). And the description is still more complex if the effects of composition are also considered. If all factors are dealt with together, the glass transition then becomes a hypersurface in the pressure–temperature–composition–timescale space.
Figure 10 Time dependence of the boundary between the glass and liquid phases of CaAl2Si2O8.
Source: Data from [32].
Figure 11 Pressure dependence of the glass transition of atactic polystyrene.
Source: Zero‐frequency Brillouin scattering data from [33].
2.3.4 An Irreversible Transition
The glass transition was first signaled by anomalous increases of the heat capacity, and its kinetic nature by the dependence of these anomalies on the thermal histories of the samples investigated (Chapter 10.11). Such effects are clearly apparent in early Cp 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 Cp is no longer defined by temperature only. The measurements made upon heating and cooling differ and Cp differences of up to 20% are found between samples initially cooled rapidly and slowly. Also noteworthy is the fact that the observed Cp 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 Cp differences between glasses cooled at different rates q; Sup. liq.: enthalpy of the equilibrium supercooled liquid.
Below 100 °C, Cp depends again only on temperature. If integrated from 270 to 100 °C, however, the Cp and Cp /T differences between the rapidly and slowly cooled samples represent enthalpy and entropy differences, respectively. These are constant below 100 °C as the glass Cp 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
