and a modifier at a CN of 6. Sun model was also specific to oxides. It did not account for chalcogenide glasses (Chapter 6.5) with typical bond strength of 170 kJ/mol along the chains (covalent bond) and less between the chains (van der Waals forces). An interesting intermediate class of oxide is formed by TeO2, MoO3, Bi2O3, Al2O3, Ga2O3, and V2O5, which do not vitrify by themselves but will do so when mixed with other (modifier) oxides.
Rawson modified Sun's criteria for glass formation [22] by considering the ratio of the bond strength and energy available at the melting point Tm instead of the coordination number. He noted that glass formation then correlates better with Eb/Tm, being achieved for values of this ratio higher than 0.05 kJ/mol K. The higher this value, the lower the probability for bonds to be broken at Tm, and hence the greater the vitrifiability. Glass formation is thus easier for high bond strength and low melting (liquidus) temperature, which implies that eutectic compositions do favor it.
Based on Rawson and Sun approaches, a modified glass‐forming criterion termed the Thermodynamic Relative Glass‐Forming Ability has recently been proposed in terms of the parameter Eb/Cp Tm, where Cp is the isobaric heat capacity [23], which can be regarded as an extension of Rawson's criteria. The ordering that results from this model is not convincing, however, and its basic ideas remain to be well justified.
Dietzel [24] characterized the ability of cations to enter the network structure by their field strength, which he defined as
(9)
where Z is its valence and r its ionic radius (Å) in the oxide. As listed in Table 2, lower field‐strength cations (e.g. alkalis) are network modifiers, whereas ions with higher field strengths (such as Si, P, or B) are network formers. Interestingly, Dietzel model is appropriate for describing phase separation, either through crystallization or unmixing, in cooled binary systems such as SiO2–P2O5, SiO2–B2O3, or B2O3–P2O5. This may generally be the case when the field strengths of two cations are approximately equal since forming a single stable crystalline compound normally requires a difference ΔF greater than 0.3. The number of possible stable compounds increases with ΔF as well as the tendency to form glass. For a binary system, glass formation is likely for Δf larger than 1.33 although this theory is useful to categorize the glass‐forming ability of conventional systems, but not universally [25].
Finally, the topological constraints theory introduced by Phillips [26] must also be mentioned. As reviewed in Chapter 2.7, it indicates that the glass‐forming tendency is maximized when the number of constraints is equal to the number of degrees of freedom in the structure.
In summary, vitrification is favored by high viscosity and configurational complexity. A more complicated chemical composition translates into a greater number of compounds that could nucleate. Owing to mutual competition between these possible crystals, nucleation and growth crystals end up being frustrated. That they do not take place upon rapid cooling thus is a consequence of a confusion principle [8].
6 Glass‐Liquid Transition
Structural theories with energetic and microstructural criteria such as topological constraints describe elements that favor glass formation, i.e. the preservation of a topologically disordered distribution of basic elements in glasses. Kinetic theory shows how to avoid crystallization rather than explaining why the vitreous state really forms through the liquid–glass transition – it is at Tg that the “drama” occurs! Although kinetically controlled, the glass transition manifests itself as a second‐order phase transformation in the sense of Ehrenfest classification. Depending on the kind of measurement performed, it is thus revealed either as a continuous change of first‐order thermodynamic properties such as volume, enthalpy, entropy, or as a discontinuous variation of second‐order thermodynamic properties such as heat capacity or thermal expansion coefficient across the glass transition range.
As indicated by its name, the CPT treats the glass transition as a percolation‐type second‐order transformation [27]. It pictures it as the disappearance in the glassy state of percolating clusters of broken bonds – configurons. Above Tg, percolating clusters, which are formed by broken bonds, enable a floppier structure and hence a greater degree of freedom for atomic motion so that it results in a higher heat capacity and thermal expansion coefficient. Below Tg there are no extended clusters of broken bonds such that the material has acquired a 3‐D structure with a bonding system similar to that of crystals except for lattice disorder. This disordered lattice then contains only point defects in the form of configurons. Agglomerates of fractal structures made of these broken bonds are present only above Tg, which is given by:
(10)
In this equation Hd and Sd are the quasi‐equilibrium (isostructural) enthalpy and entropy of configurons present in Eq. (7) and ϕc is the percolation threshold, i.e. the critical fraction of space occupied by spheres of bond‐length diameters located within the bonding sites of the disordered lattice.
For strong melts such as SiO2, the percolation threshold in Eq. (10) is given by the theoretical (universal) Scher–Zallen critical density ϕc of 0.15 ± 0.01, which results in a practical coincidence between the calculated and measured Tg values. The parameter Hm has no influence on Tg as it characterizes the mobility of atoms or molecules through the high‐temperature fluidity of the melt – see Eq. (7). Because Hd is half of bond strength (Table 2), Eq. (10) shows that the higher this strength, the higher Tg. The vacancy model of the generalized lattice theory of associated solutions provides direct means to calculate thermodynamic properties as well as the relative number of bonds formed in glasses and melts when the second coordination sphere of atoms is taken into consideration [28].
In terms of chemical bonds, an amorphous material transforms to a glass on cooling when the topology of connections changes (Table 3), i.e. when the Hausdorff dimensionality of broken bonds changes from the 2.5 value of a fractal percolating cluster made of broken bonds to the zero value of a 3‐D solid. In terms of bonding lattice, the transition from the glass to the liquid upon heating may be explained as a reduction of the topological signature (i.e. Hausdorff dimensionality [29]) of the disordered bonding lattice from 3 for a glass (3‐D bonded material) to the fractal Df of 2.4–2.8 of the melt. These are the main changes that account for the drastic variations in material properties at glass‐to‐liquid transition [27].
Table 2 Classification of cations according to Diezel's field strength.
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