Читать книгу Encyclopedia of Glass Science, Technology, History, and Culture - Группа авторов - Страница 282

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.