Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
Therefore, the number of independent constraints per vertex (nu = Nu/V) in a rigid unit is
(4)
A major advantage of PCT is that Eq. (3) counts correctly the number of independent constraints in a rigid unit. From Eq. (4) and the values of nu listed in Table 1 for several simple polyhedral units, one sees that nu increases with both V (for fixed δ) and δ (for fixed V > δ). When a structural unit is non‐regular and rigid, other parameters, in addition to δ and V, are needed to specify the structural unit.
3.2 Existence of Topologically Disordered (d = 3) Networks
For an extended three‐dimensional network (made up of a single type of structural unit) with an average C structural units sharing a vertex, the degrees of freedom, f, per vertex are
(5)
If f is positive, a network can exist. When f is negative, a TD network cannot exist. Thus, f = 0 provides a boundary for the existence of TD networks.
If additional constraints (θ) are present at the shared corners (for example, bond angle constraints) or if there are internal degrees of freedom (h) within the structural units (for example, there is one internal degree of freedom in a unit made up of a pair of edge shared tetrahedra), then Eq. (5) can be modified as follows:
(6)
The degrees of freedom of TD networks are also listed in Table 1 for several rigid structural units for different values of connectivity. It should be noted that SiO2 with V = 4, C = 2, δ = d = 3 satisfies the condition of isostaticity (f = 0). Similarly, a two‐dimensional TD network of corner‐sharing rigid triangles (a candidate structure of B2O3 glass) is also isostatic.
3.3 Glass‐forming Ability
According to PCT, a glass can be formed if and only if it can exist as a TD network (i.e. only if f ≥ 0). With increasingly positive values of f, however, the existing TD network becomes progressively more floppy and may crystallize beyond a certain value f(q) that depends on the cooling rate, q. Thus, glass formation is possible in the range for which 0 ≤ f ≤ f(q). With positive and increasing f, the potential energy of interaction (the chemical energy) increases because of unsatisfied chemical bonds. With decreasing and negative f, the strain energy in the system increases. In other words, an isostatic network represents a minimum in the total energy of the system. For this reason, the glass‐forming ability is best under isostatic condition (f = 0) and becomes poor as f increases. The f = 0 boundary is termed the isostatic boundary of glass formation and the f(q) boundary the kinetic boundary of glass formation.
3.3.1 Glass‐forming Ability and the Condition of Isostaticity (f = 0)
The isostaticity condition is satisfied in three dimensions for tetrahedral structural units (V = 4) with two units sharing every vertex (C = 2) as is the case for SiO2, GeO2, and BeF2, which are known as excellent glass formers. The isostatic condition is also satisfied for two‐dimensional networks made of corner‐sharing triangles. This is often considered to be the reason why B2O3 is a strong glass former.
An interesting application of the isostatic boundary concept is identification of limiting isostatic composition for glass formation. Consider the example of nitridation of alkali‐silicate glasses. In silicon oxynitride glasses, nitrogen substitutes for oxygen forming two kinds of vertices: oxygen vertices with C = 2 and nitrogen vertices with C = 3. Adding nitrogen to silica (for which f is 0) makes f negative. This suggests that nitridation of silica will be difficult. However, addition of alkali creates non‐bridging oxygens (with C = 1). Thus, nitrogen can be added to alkali‐silicates while keeping f non‐negative. In fact one can calculate the maximum amount of nitrogen that can be incorporated into an alkali‐silicate glass as a function of the alkali content. Consider glass formation in an alkali silicon oxynitride system of the general composition x Na2O·(1 − x)[SiO(2−y) N(2y/3)]. Note that 0 ≤ y ≤ 2 and 0 ≤ x ≤ 1. This system has three types of vertices: non‐bridging oxygens with C = 1, bridging oxygens with C = 2, and bridging nitrogens with C = 3. The isostatic condition gives the limiting solubility of nitrogen, ymax = 3x/(1 − x). For y > ymax, f becomes negative. Whereas systematic investigations of nitridation of alkali‐silicate glasses are not available, it is known that nitridation becomes easier upon increasing the alkali content [16].
Another example is provided by binary alkali‐tellurite systems. Pure TeO2 with trigonal bipyramid structural units is over‐constrained and does not form glass. Glass formation improves upon addition of alkali oxide because of formation of non‐bridging oxygens, thereby increasing f and thus making it possible to form glasses when sufficient alkali oxide is added. Narayanan and Zwanziger [17] have rationalized in this way glass formation in alkali‐tellurite systems.
3.3.2 Glass Formation Under Hypostatic (f > 0) Conditions
This condition is best exemplified by the x Na2O·(1 − x) SiO2 system where addition of Na2O leads to conversion of bridging oxygens (with C = 2) into non‐bridging oxygens (with C = 1). The average value of degrees of freedom increases with increase in x:
(7)
It is well known that glass formation in alkali‐silicate systems becomes difficult for large value of x, especially when x > 0.5.
3.3.3 Glass Formation Under Hyperstatic (f < 0) Conditions
When f < 0, a TD network cannot exist. The excess strain energy can, however, be accommodated in a variety of ways that increase the value of f toward f = 0. One possibility is that the network crystallizes, thereby reducing the number of independent constraints by converting some into dependent ones. A second possibility