The reason for this stability is that relaxation processes are controlled by viscosity. The characteristic time τ required to achieve time‐independent parameters can be derived from Maxwell's simple relaxation model, which predicts that τ = η/G, where G is the shear modulus and η the viscosity (Chapter 3.7). The higher the viscosity, the longer the relaxation time. Besides, viscosity changes are thermally activated. Glass‐forming oxides are characterized by high activation energies and very high viscosities under normal conditions. As an extreme example, fused silica has an activation energy for viscous flow of QH of 759 kJ/mol at lower temperatures and a shear modus of about 31 GPa, which implies relaxation times τM as long as 1098 years at room temperature, an immeasurably longer time than even the 14∙109 years lifetime of the universe.
3 Kinetic Theory of Vitrification
As already stated, vitrification on cooling is tantamount to bypassing crystallization, which would bring the material to an ordered state of matter of lower energy and, thus, of greater stability. If cooled quickly enough, virtually any material can vitrify. Hence, an important question is to know how fast the melt should be cooled to avoid detectable crystallization. Actually, the cooling rates of good glass formers range from 1 to 10 K/s and those of poor ones are higher than ~100 K/s. Familiar glass‐forming materials such as SiO2‐rich silicates thus do not require fast cooling. An extreme example is supercooled NaAlSi3O8, which did not crystallize at all after five years spent 90 K below the melting point of the albite feldspar mineral [4]. Even though the quenching rate of a liquid medium can be increased to about 103 K/s, such high values may not be fast enough to avoid partial or complete crystallization. In silicates, SiO2‐poor olivine compositions (e.g. Mg2SiO4) are notorious examples of liquids that are difficult to vitrify at such rates [5]. Quickly crystallizing liquids such as metals may require still faster cooling. For instance, early metallic glasses (Chapter 7.10) had to be splat quenched in the form of thin ribbons at about 106 K/s to avoid crystallization, mainly because their viscosities remain very low with decreasing temperatures (Figure 1).
An added complication is that melts can be good glass formers only within limited composition regions as indicated in Figure 2 for aluminosilicates. Compilations of glass‐forming regions of phase diagrams are available [6–8]. As a rule, however, glass formation is enhanced in more complex systems. Oxide systems containing a variety of cations are easier to obtain in a glassy state as their liquidus temperature is lower and their complexity necessitates longer times for diffusion‐controlled redistribution of their diverse constituents before crystal nucleation and growth can take place.
Figure 1 Critical cooling rates for glass formation. Reduced glass transition temperature Trg defined as Tg/Tm, where Tm is the liquidus temperature.
Figure 2 Glass formation ranges in aluminosilicate systems (shaded areas).
Source: After [6], courtesy P. Richet.
If crystallization is bypassed, the melt undergoes the glass transition in a temperature range that shifts to higher temperatures for higher cooling rates. Both the width of this range and the value of Tg within it may vary by typically several tens of degrees. As the width and variation are small compared with Tg, it is therefore possible to manage a cooling schedule rapid enough to keep the degree of crystallization negligible. Volume fractions considered to be negligible are lower than 1 ppm, which is the typical instrumental limit for detecting the presence of crystals by microanalytical techniques (Chapter 2.3).
If v designates the volume fraction of crystalline(s) phase(s), the material is amorphous if v = 0, crystalline if v = 1, and polyphase or heterogeneous when 0 < v < 1. Under isothermal conditions the volume fraction of a growing crystalline phase varies with time as described by the Johnson–Mehl–Avrami–Kolmogorov equation (see [9] for references and details):
(1)
where u is the rate of crystal growth and Iv the nucleation rate.
If both rates can be estimated, the volume fraction of the crystalline phase v achieved for a given cooling rate can be calculated and the results be compared with experimental data. The rate of homogeneous nucleation is given by James equation:
(2)
Here W* is the thermodynamic barrier to homogeneous nucleation, nv the number of molecules or formula units of nucleating phase per unit volume, λ a jump distance, and η viscosity. For heterogeneous nucleation, the thermodynamic barrier to nucleation actually becomes Wh* = W*(2 + cosθ)(1 − cosθ)2/4, where θ is the contact angle between the crystal and the nucleating heterogeneity. The rate of crystal growth is given by Wilson–Frenkel equation:
(3)
where f is the interface site factor, Dc the kinetic (diffusion) coefficient, Vm the molar volume, and ΔGv the difference in Gibbs free energy between unit volumes of the crystal and liquid.
One then obtains the critical cooling time (tc) and rate (qc), for a defined volume fraction of crystals vc. Taking the aforementioned value vc of 1 ppm, one obtains
Figure 3 Determination of the critical cooling rate from a time temperature transformation diagram.
(4a)
and
(4b)
Despite its general correctness and qualitative agreement with experiments, the kinetic theory suffers from limited quantitative applications. Whereas quantitative agreement has been achieved for simple