7). If only high‐temperature measurements are considered, then a simpler Arrhenius equation is generally adequate, viz.
(4)
where η0 is a pre‐exponential term and ∆Hη the activation enthalpy for viscous flow. Consistent with the aforementioned effects of thermal history (Figure 6b), the increasing departure of the viscosities from an Arrhenius fit made to the high‐temperature data (Figure 7) indicates that, independently of any thermal‐energy decrease, the structural rearrangements induced by lower temperatures progressively hinders viscous flow. The effect is still more apparent when measurements are made rapidly such that structural relaxation does not take place. Under these conditions, the isoconfigurational viscosity is indeed lower than the viscosity of the equilibrium supercooled liquid at the same temperature (Figure 7).
Figure 7 Viscosity of window glass; solid line VFT fit to the data; dashed line: Arrhenius fit made to the high‐temperature measurements; arrow: onset of departure from the equilibrium viscosity; solid squares and line: isostructural viscosities.
Source: Data from [26, 27].
2.3 The Glass Transition
2.3.1 Standard Glass‐Transition Temperature
For the experimental timescales of the order of a few minutes typical of measurements of macroscopic properties, one observes that, regardless of chemical composition, time‐dependent results begin to be observed when the viscosity becomes higher than about 1012 Pa. For convenience and comparison purposes, one defines the standard glass‐transition temperature, Tg, as the temperature at which the viscosity of the liquid reaches this value of 1012 Pa.s.
2.3.2 Volume Effects
The enhanced thermal expansion coefficient observed upon heating of a glass rod in dilatometry experiments is one of the most familiar manifestations of the glass transition (Figure 8a). The marked increase over an interval of about 50 K is rapidly followed by sample collapse because the viscosity rapidly decreases so much that the sample begins to flow under its own weight before structural relaxation is complete. As a result, the volume thermal expansion coefficient [α = 1/V (∂V/∂T)P = 3/l (∂l/∂T)P] may be rigorously determined from the slope of the dilatometry curve for the glass, but not for the supercooled liquid.
Figure 8 Volume effects of the glass transition. (a) Linear thermal expansion coefficient of E glass (Chapter 1.6) heated at 10 K/min; l = sample length
(Source: Data from [28]).
(b). Dependence of the volume of a glass on its fictive temperature.
In dilatometry experiments, one usually defines the glass‐transition temperature as the intersection of the tangents to the lower‐ and higher‐temperature curves. This temperature generally differs somewhat from the standard Tg simply because the glass transition depends on the particular experimental conditions of the experiment. With respect to enthalpimetry, dilatometry has the advantage of yielding absolute values of the property of interest, namely, the volume (and density). The influence of thermal history on density can thus be readily determined (which is why it was observed as early as in 1845, cf. Chapter 10.11). In contrast, the thermal expansion coefficient of glasses generally does not markedly depend on thermal history. At least above room temperature. The volumes of glasses produced at different cooling rates will then plot as a series of parallel lines (Figure 8b). To characterize the state of the glass, it suffices to know the temperature at which equilibrium was lost, which is directly given by the intersection of the glass and supercooled volumes (Figure 8b). This parameter is called the fictive temperature (
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
