polyhedral structural units and by the vertex‐connectivity (C) that specifies the number of polyhedral units sharing a vertex. The vertex‐connectivity is related to (see Eq. 1) but is different from the average coordination number r.
3.1 Rigidity of Polyhedral Structural Units
An isolated single regular polyhedral unit can be specified by two parameters: the dimension (δ) of the unit and its number of vertices (V). The dimension of the unit is the minimum dimension necessary to embed it. For example, δ = 1 for a rod, 2 for a triangular unit, and 3 for a tetrahedron. It is clear that δ ≤ d (the dimension of the network) and that V ≥ (δ + 1).
When a regular polyhedral structural unit is rigid, the total number, Nu, of independent constraints in the unit satisfies the following relation:
(3)
Table 1 Degrees of freedom (f) of d‐dimensional TD networks of rigid units (δ, V) with C units sharing a vertex (with the assumption h = θ = 0) based on Eqs. (4) and (5).
| Structural unit | δ | V | n u | d | C | f |
|---|---|---|---|---|---|---|
| Rod | 1 | 2 | 0.5 | 2 | 3 | 0.5 |
| 3 | 4 | 1 | ||||
| 3 | 6 | 0 | ||||
| Triangle | 2 | 3 | 1 | 2 | 2 | 0 |
| 3 | 2 | 1 | ||||
| Square | 2 | 4 | 1.25 | 2 | 2 | −0.5 |
| Tetrahedron | 3 | 4 | 1.5 | 3 | 2 | 0 |
| 3 | 3 | −1.5 | ||||
| Octahedron | 3 | 6 | 2 | 3 | 2 | −1 |
| Cube | 3 | 8 | 2.25 | 3 | 2 | −1.5 |
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
