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 is that new structural units form by breaking weaker constraints. When constraints are broken within structural units, the polyhedra become distorted. The existence of an extended TD network of distorted ABV polyhedral units (where the BAB angular constraint is broken at the A site but the AB length constraint remains intact) is demonstrated by the example of glass formation in the CaO–Al2O3 binary system. Since neither component is a network former, glass formation in this binary system is poor. If the presence of CaO stabilizes four‐coordinated aluminum ions, AlIV, with two AlO4 tetrahedra sharing an oxygen vertex (as in silica), then the composition having 50% CaO should be a good glass former. However, experimental results show that glasses in this system form only in a small composition range at about 65% CaO [18]. It is possible to rationalize this observation with the constraint theory. Recently, Jahn and Madden [19] have reported from MD simulations that at 2350 K aluminum is present in Al2O3 melt in several different coordination states; about 54% AlIV, 41% AlV, 4% AlVI, and 1% AlIII. Further, it was observed that some of the oxygens are present as OIII, oxygen coordinated by three aluminums (also known as oxygen triclusters), and the remaining as normal bridging oxygens, OII. One can use this structural information and PCT to rationalize why the 65% CaO composition forms best glasses in this system. First, it can be assumed that the structures of Ca‐aluminate and alumina melts are similar except for the incorporation into the network of O from CaO. Next, one can simplify the structural information by neglecting the concentrations of AlVI and AlIII. Let z be the fraction of AlIV and (1 − z) that of AlV. Then, for the composition x CaO·(1 − x)Al2O3, one shows with PCT that the degrees of freedom, f, is given by
(8)
When f = 0, Eq. (8) gives the following expression for the isostatic composition x* [x(f = 0)] in terms of z:
(9)
For z = 0.54, Eq. (9) gives x* = 0.65, the same value as for the best glass‐forming composition [18].
3.4 Existence of Super‐Structural Units
In B2O3 glass the presence of rigid, planar boroxol B3O6 units made up of three trigonal BO3 units is well established [20]. In contrast to basic polyhedral units, ABV, where a single A atom is coordinated by B atoms, super‐structural units such as boroxol units contain more than one A atom. Do super‐structural units then exist in other borate glasses? It is a question that has long persisted in the oxide‐glass science.
Super‐structural units may be energetically more favorable in systems with long‐range interactions. However, their larger size raises difficulties to match the density of networks with the observed density of glass. For example, the boroxol units are topologically equivalent to the basic BO3 trigonal unit (both have δ = 2 and V = 3). If all BO3 trigonal units in B2O3 glass are replaced
