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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 T_rg defined as T_g/T_m, where T_m 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 T_g within it may vary by typically several tens of degrees. As the width and variation are small compared with T_g, 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 I_v 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, n_v 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 W_h* = 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, D_c the kinetic (diffusion) coefficient, V_m the molar volume, and ΔG_v the difference in Gibbs free energy between unit volumes of the crystal and liquid.

One then obtains the critical cooling time (t_c) and rate (q_c), for a defined volume fraction of crystals v_c. Taking the aforementioned value v_c 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 silicate systems, the discrepancy is of many orders of magnitude in most cases [6, 8].

In practice, the critical cooling rate (CCR) is determined from the so‐called time temperature transformation (TTT) diagrams, which represent nose‐shaped curves with a constant crystal fraction on time–temperature axes (Figure 3). To ensure the obtention of a glass with a crystal volume fraction lower than v, it is required to follow a cooling pathway such that the cooling line (curve) will not touch the nose [10]. The CCR is then found as follows:

(5)

Examples are listed in Table 1.