Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

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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 T_g 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 T_g, 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 T_g 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 T_g, which is given by:

(10)

      In this equation H_d and S_d are the quasi‐equilibrium (isostructural) enthalpy and entropy of configurons present in Eq. (7) and ϕ_c 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 ϕ_c of 0.15 ± 0.01, which results in a practical coincidence between the calculated and measured T_g values. The parameter H_m has no influence on T_g as it characterizes the mobility of atoms or molecules through the high‐temperature fluidity of the melt – see Eq. (7). Because H_d is half of bond strength (Table 2), Eq. (10) shows that the higher this strength, the higher T_g. 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 D_f 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.

Element	Valence Z	Ionic distance for oxides, Å	Coordination number	Field strength, 1/Å2	Bond strength, kJ/mol	Function
Si	4	1.60	4	1.57	443	Network formers: F~1.5–2.0
B	3	1.50	3	1.63	498
	4		4	1.34	372
P	5	1.55	4	2.1	368–464
Ti	4	1.96	4	1.25	455	Intermediates: F~0.5–1.0
	4	1.96	6	1.04	304
Al	3	1.77	4	0.96	335–423
	3	1.89	6	0.84	224–284
Fe	3	1.88	4	0.85
	3	1.99	6	0.76
Be	2	1.53	4	0.86	263
Zr	4		6	0.84	338
	4	2.28	8	0.77	255
Mg	2	2.03	4	0.53
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