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 by RD values higher and lower than 2, respectively.
Many equations have been proposed to express viscosity–temperature relationships (Chapter 4.1, [13–15]). Consistent with Doremus criterion, an exponential expression with activation energies QL and QH at low and high temperatures, respectively, [13] will be used here because it results from the configuron percolation theory (CPT), which accounts for viscous flow in terms of elementary excitations resulting from broken bonds named configurons [14, 16]. This equation (Sheffield model) is
(7)
where A1 = k/6πrD0, A2 = exp(−Sm/R), B = Hm, C = exp(−Sd/R) and D = Hd, R the gas constant, k Boltzmann constant, r the configuron radius D0 = fgλ 2 zp0 ν0, f the correlation factor, g a geometrical factor (~1/6), λ the average jump length, ν0 the configuron vibrational frequency or the frequency (with which the configuron attempts to surmount the energy barrier to jump into a neighboring site), z the number of nearest neighbors, p0 a configuration factor, Hd the enthalpy, Sd the entropy of formation, and Hm and Sm the enthalpy and entropy of motion of the configurons.
In practice, one finds that A2 exp(B/RT) > > 1, i.e. that four parameters usually suffice for Eq. (7) to be fitted to practically all available experimental viscosity data [17]. Comparison with other viscosity models and numerical calculations have confirmed the excellent description of the viscosity provided by Eq. (7) for simple and complex organic and inorganic compositions (e.g. [13]). This equation can be readily approximated within narrow temperature intervals by expressions derived from the well‐known Vogel‐Tammann‐Fulcher, Adam–Gibbs, Avramov–Milchev, or Sanditov models [14, 15, 17]. It can be used at all temperatures and gives the correct Arrhenius‐type asymptotes at high and low temperatures, namely η(T) ~ exp(QH/RT) at T << Tg, and η(T) ~ exp(QL/RT) at T >> Tg, where QH = Hd + Hm and QL = Hm. Obviously, the activation energy of viscosity reduces to a low value equal to Hm at high temperatures when temperature fluctuations create plenty of configurons. In contrast, some bonds need to be broken in the glassy state as temperature fluctuations do not create them effectively so that the activation energy then takes its full value QH = Hd + Hm.
5 Structural Factors
Apart from inhomogeneities and potential phase separation, glasses lack long‐range order but do possess short‐ and medium‐range ordering (Chapter 2.1). A number of models have aimed at revealing the most characteristic structural aspects of good