Figure 1 Deformation of a finite network of four nodes. Lines represent linear constraints. Top: floppy network (hypostatic, f > 0) with only four constraints originating from the edges. Middle: addition of a diagonal constraint makes the network rigid (isostatic with f = 0). Bottom: addition of a second, dependent diagonal constraint does not change the rigidity of the network.
2.4 Degrees of Freedom and the Network Deformation Modes
Without constraints (as, for example, in an ideal gas), each atom has d coordinate degrees of freedom where d is the dimension of the network. As constraints are added at a vertex, its coordinate degrees of freedom decrease. If n is the average number of independent constraints per vertex, then the average degrees of freedom per vertex, f, in a network is given by
(2)
If n < d, then f is positive and the network can deform without expenditure of energy. Such a network is termed “floppy” (or hypostatic) and has exactly f floppy (soft or low frequency) modes per vertex. The number of degrees of freedom decreases as n increases. When n > d, the network is over‐constrained and is termed “stressed‐rigid” (or hyperstatic). The excess (n − d) constraints in a stressed‐rigid network are dependent if such a network exists. The transition from floppy to stressed‐rigid takes place at n = d (i.e. at f = 0), which marks the disappearance of floppy modes and the onset of network rigidity. Such a network is called isostatic (Figure 1).
In a floppy network, there may exist finite‐size rigid inclusions (small group of atoms interconnected in a rigid manner) that are embedded in a floppy matrix. The average size of such rigid clusters grows as n increases till the rigid clusters begin to percolate, causing a transition from a floppy into a rigid network at n = d. Similarly, when a network is stressed‐rigid (n > d), it may contain floppy clusters in a rigid matrix. The average size of these floppy clusters grows as n is reduced so that at n = d, the floppy clusters begin to percolate making the entire network floppy. Thus, a network undergoes a rigidity percolation transition at f = 0. This basic idea is at the heart of most TCT applications because n can vary with changes in both temperature and composition. In other words, since n = n(T, x), the isostatic boundary in a T–x phase diagram is described by n(T, x) = d.
3 Polyhedral Constraint Theory
As mentioned before, PCT considers only chemically ordered networks which, according to Zachariasen [2], are TD networks of structural units made up of corner‐sharing rigid polyhedra. In such networks, it is convenient to treat the shared corners of the polyhedra as the vertices and the polyhedral structural units as links. The rigidity of a network then arises from the rigidity of the structural units as well as from the vertex‐connectivity condition (i.e. the fact that all corners of polyhedral units are shared among a certain number of units). These constraints resulting from the rigidity of structural units and connectivity are termed polyhedral constraints.
The deformation of a TD network is determined by the type of 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 |
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