have been proposed to account for them, and to highlight some simplifying features thanks to which intrinsically complex glass problems become more tractable.
A main source of difficulty is that the time parameter must be considered because of the kinetic nature of the glass transition. In the backdrop is the way in which the Gibbs free energy of a glass‐forming liquid would be minimized under given experimental conditions and, thus, the kinetics at which physical properties relax after changes in intensive thermodynamic variables (Chapter 3.7). The largest and most rapid decrease of the Gibbs free energy would of course be ensured by crystallization. To bypass it, it has been known from time immemorial that a melt must be cooled rapidly enough. Other things being equal, vitrification is favored by large freezing‐point depressions near eutectic compositions, which result in increased viscosities and reduced thermodynamic driving forces for crystallization.
With very few exceptions (e.g. [25]), however, supercooled liquids do crystallize more or less rapidly upon prolonged annealing. Perhaps also influenced by the early twentieth‐century conception that glasses were supercooled liquids (Chapter 10.11), a commonly held assumption is that any glass would eventually crystallize. This assumption is in fact plainly contradicted by the 4.6‐billion year old glasses found in meteorites (Chapter 7.1). What has ensured their long‐term preservation has been the extremely dry conditions of extraterrestrial space, which have prevented them from weathering. Since their SiO2‐poor compositions would make them prime candidates for ready devitrification, the almost infinite metastability enjoyed by these glasses is especially significant. The crystallization issue will thus be left aside in the following.
2.2 Relaxation: Phenomenological Aspects
Atomic mobility is the hallmark of the molten state as illustrated by the ready flow of a liquid adjusting to the shape of its container. Contrary to crystals where atomic positions are fixed and strongly constrained by long‐range symmetry, liquids are characterized by dynamic disorder, i.e. by unceasing atomic rearrangements. This structural incompatibility between a crystal and a liquid makes any progressive transformation of one phase into the other impossible. In contrast, the vitrification of liquids is clearly a continuous process during which disordered structures become frozen in as revealed by progressively increasing viscosities, which eventually becomes so high that the materials have mechanically become a solid.
At high temperatures, the liquid is in internal thermodynamic equilibrium because its properties are time independent and uniquely determined by two intensive variables, usually taken to be pressure and temperature. At high viscosities, however, this simplicity no longer holds true as seen if one exerts a stress on the liquid at constant temperature or change the temperature at constant stress (Figure 6a). For a window glass [26], a constant, equilibrium shear viscosity is, for example, reached more rapidly in the former case than in the latter but this difference does not need to be commented upon here because pressure and temperature changes are of a different nature. Of greater importance is that Boltzmann superposition principle (Chapter 10.11) applies because, if both perturbations are simultaneously exerted, the response of the system is the sum of the two individual responses (Figure 6a).
Figure 6 Viscosity relaxation of window glass
(Source: Data from [26]).
(a) Time dependence of the viscosity at 788 K after: (1) application of a 110 MPa stress; (2) a temperature change from 819 to 788 K with this stress; (3) exerting simultaneously these stress and temperature changes. (b) Attainment of the equilibrium viscosity; sample equilibrated at 795 K, then quickly brought for equilibration at 788 K and at 777 K (open symbols) before following the same procedure for reversing the equilibrium values first measured at 788 and 795 K (open symbols).
In practice, temperature changes matter most. When high viscosities are measured at successively lower temperatures and then at higher temperatures (Figure 6b), two conclusions follow: (i) the time needed to reach the constant equilibrium values increases tremendously with decreasing temperatures; (ii) the approach to equilibrium is slower when the sample was previously equilibrated at a lower than at a higher temperature. Hence, the rate at which these changes occur depends not only on temperature but also on the thermal history of the sample, i.e. on the instantaneous structure as well. Because thermodynamic equilibrium is reached when the structure has adjusted to the new intensive parameters, the process is termed structural relaxation.
To characterize the rate at which the shear viscosity (η) or any other property Y approaches a new equilibrium value, Ye, one defines the relaxation time, τY, as
(1)
where Yt is the value actually measured at time t. If τY were constant, the relaxation would be exponential:
(2)
where Y0 is the initial Y value, so that after a time τY, the variation of Y would be a fraction 1/e of the initial departure from the equilibrium value. Regardless of the actual non‐exponential nature of relaxation, measurements, for example, made on window glass at 777 K point to relaxation times much higher than one hour (Figure 6b). A measurement performed in only a few minutes would thus refer to a fixed configuration, i.e. to a glass. Depending on the timescale of the experiment, one observes that the nature of response is thus either liquid‐ or solid‐like.
The glass transition range is that temperature interval where, depending on the timescale of the experiment performed, time‐dependent observations are made. It signals the change from the liquid state, where a great many different atomic configurations are unceasingly explored, to another state where atoms become trapped in fixed positions and properties become again time independent. In statistical–mechanical jargon, this change is said to represent the loss of ergodicity and, thus, of internal thermodynamic equilibrium.
Experimentally, the loss of equilibrium can be readily followed by viscometry. Over an interval as wide as 10–1015.5 Pa.s, the viscosity of a glass‐forming melt can be reproduced empirically with the Vogel–Fulcher–Tammann (VFT) equation (Chapters 4.1 and 10.11):
(3)
where A, B, and T1 are constants
