2.3.3 Frequency Dependence
For exploring further the kinetics of the glass transition, one can vary the experimental timescale not only through changes of the heating rate for a given technique, but through changes of the technique itself. In view of their relative simplicity, acoustic measurements of the adiabatic compressibility are especially interesting in this respect. For an isotropic solid, this compressibility is related to the velocities of compressional (vp) and transverse (vs) acoustic waves by:
(5)
where ρ is the density. In a liquid of low viscosity, the attenuation of compressional waves is so rapid that one can usually consider that these waves do not propagate at all, in which case the compressibility reduces to
(6)
Acoustic measurements are typically made with transducers working at MHz frequencies. Under these conditions, the response of the material to the compression exerted adiabatically by the acoustic waves is probed at timescales of the order of 10−6 seconds. To be induced by an acoustic wave, configurational changes must thus take place at timescales at least 106–107 shorter than those of dilatometry or calorimetry experiments. Their onset is thus correlatively observed at much higher temperatures. For a sodium silicate (Figure 9a), they are revealed above 700 °C by a temperature interval where vp decreases markedly and becomes frequency‐dependent. With respect to dilatometry or calorimetry experiments, the glass transition shifts from about 500 to 900 °C, with a difference of about 50° between the measurements made at 1 and 5.6 MHz. At higher temperatures, equilibrium values of the compressibility are finally measured 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