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 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.
4 The Viscosity Factor
The definition of the glass transition in terms of the standard glass transition derived from the viscosity–temperature relationship, η(T), is not thermodynamic, but operational [8]. As determined from changes in second‐order derivatives of thermodynamic variables, the glass transition takes place at viscosities in the range 108–1013 Pa s depending on the cooling rate. In view of the very steep temperature dependence of viscosity, a great difference in this parameter results in only small variations of the operational Tg.
Table 1 Critical cooling rate for some glasses, K/s.
Source: After [9].
| Material | Nucleation mechanism | |||
|---|---|---|---|---|
| Homogeneous | Heterogeneous θ = 100° | Heterogeneous θ = 60° | Heterogeneous θ = 40° | |
| SiO2 | 9 × 10−6 | 10−5 | 8 × 10−3 | 2 × 10−1 |
| GeO2 | 3 × 10−3 | 3 × 10−3 | 1 | 20 |
| Na2O 2 SiO2 | 6 × 10−3 | 8 × 10−3 | 10 | 3 × 102 |
| Salol [C13H10O3] | 10 | |||
| Water | 107 | |||
| Ag | 1010 |
As has long been recognized, glass formation is therefore easier in eutectic regions because freezing‐point depressions enable lower temperatures and higher viscosities to be reached [11]. However, viscosity at the liquidus is not a single scaling parameter for assessing glass‐forming ability. The temperature dependence of the viscosity below Tm must also be considered because vitrification takes place much below the liquidus [6, 8].
Viscous flow in glass‐forming liquids is characterized by deviations from Arrhenius laws with an activation energy Q that decreases from a high QH near the glass transition to a low QL at superliquidus temperatures. As a fragility index characterizing the temperature dependence of viscosity, Doremus [12] has proposed the ratio:
(6)
Short (or fragile) and long (or strong) glass melts are, therefore, characterized
