structural unit is doubled (the volume thus increasing by a factor of 8) while the mass of the unit is only tripled so that the density of a network of boroxol units is only 3/8 of that of a network of trigonal units. Further, based on topological considerations mentioned before, an extended 3‐D network can incorporate only a small fraction of super‐structural units with V > 4. For this reason, the di‐pentaborate and the di‐triborate groups, two of the six super‐structural units listed by Wright [20], probably do not exist in significant concentrations.
4 The Bond Constraint Theory
As originally formulated by Phillips [1] for covalent networks, structural units are not considered in BCT. Instead, the system is viewed as a network of atoms at the vertices and covalent linear bonds at the edges. These covalent linear bonds provide ri/2 linear constraints at the ith vertex of coordination number ri. In addition, there also exist [ri (d − 1) −d(d − 1)/2] covalent angular‐bond constraints at the ith vertex for a d‐dimensional network. The average number of constraints, n, per vertex is, therefore,
(10)
where r is the average vertex coordination number. The condition of isostaticity (n = d) gives the following value for the critical coordination number r* (also called the rigidity percolation threshold):
(11)
Note that r* = 2 for d = 2 and r* = 2.4 for d = 3. It must be emphasized that Eqs. (10) and (11) assume that the angular constraints are intact at every vertex. This assumption does not always hold true as illustrated by silica where the angular constraints at oxygens are broken, which is generally the case for elements that do not belong to groups III, IV, and V and do not exhibit sp(n) hybridization.
Application of BCT to non‐covalent systems with long‐range interactions such as ionic systems is approximate at best, and questionable most of the time, because these systems do not lend themselves to the count of simple nearest‐neighbor constraint. For ionic systems, it is thus preferable to use PCT with structural units defined by the radius ratio of cations to anions.
4.1 Self‐organization and the Intermediate Phase
Self‐organization designates chemical and/or topological rearrangements in a network that take place spontaneously to reduce the overall energy in the system [21]. An important consequence of this process is that it allows a system to exist as an isostatic network over a range of coordination numbers or compositions. This range is sometimes known as the intermediate phase or reversibility window [22]. The range depends on the system considered and, to some extent, on its thermal history as well as on the property being measured (e.g. enthalpy release during relaxation, Raman frequency shifts in glasses, or activation energy of viscosity). For example, a coordination number range from 2.39 to about 2.52 has been reported for the intermediate phase in GexSe(1−x) system from Raman frequency shifts [22], and a range from r = 2.35 to about 2.45 in the (Na2O)x(SiO2)(1−x) system from enthalpy relaxation [23]. Interestingly, Wang et al. [24] found no evidence of any intermediate phase in the Ge–As–Se system. Also, Shatnawi et al. [25] found no discontinuities or breaks but only smooth variation with respect to composition in the structural response of GexSe(1−x) glasses in the range 0.15 < x < 0.40 implying the absence of any phase transition associated with the start and end of the intermediate phase range.
4.2 Non‐bridging Vertices (or Singly Coordinated Atoms)
There has been much discussion in the literature [26] about the role of dangling vertices (or non‐bridging nodes) and their influence (if any) on the rigidity characteristics of a network. At least conceptually, it is clear that dangling vertices should not affect the stiffness of the network because they are not network‐forming. In this respect, a confusion in the literature exists primarily because of the way constraints and degrees of freedom are counted. Clearly, if a dangling vertex is counted as being part of the network, then it is necessary to count also the length and angle constraints associated with it. A dangling vertex adds three degrees of freedom but also three constraints (one length and two angles), so that it does not make any net contribution to the degrees of freedom in a network if the counting is done correctly. However, the problem is that extra degrees of freedom often appear when the angular constraints of the dangling vertices are not included in the count [26, 27]. Because these extra degrees of freedom are associated with the floppiness of the dangling vertices themselves, they do not influence the rigidity or the flexibility of the underlying network. Thus, the opinion of this writer is that it is best to disregard the onefold coordinated atoms (i.e. dangling or non‐bridging vertices) as they have no influence on the rigidity characteristics of a network.
4.3 Glass‐forming Ability in Chalcogenide Systems
4.3.1 Ge–Se System
For the binary GexSe(1−x) system, the average coordination number r(x) is (2 + 2x) since r(Ge) = 4 and r(Se) = 2, and the system is isostatic at x* = 0.2 (corresponding to the Ge1/5 Se4/5 composition). According to Tichy and Ticha [28], however, the best glass‐forming compositions lie on the Se‐rich side in the range 0.06 < x < 0.15. To rationalize this apparent discrepancy, Tichy and Ticha [28] suggested to view the GexSe(1−x) system as an extended chemically ordered network of short, linearly‐rigid (Se)k chains containing k seleniums that are cross‐linked by Ge atoms. The ends of the Se chains are connected to Ge atoms so that four such chains share a Ge atom. Note that since x = 1/(1 + 2 k), k is 2 at x = 0.2. A network with k > 2 (corresponding to x < 0.2) should be floppy (f > 0). However, one can show that some of the degrees of freedom (let us denote these by f # ) in such a network for x < 0.2 are associated with the dihedral rotation of the inner seleniums (Se#), those inside the selenium chains (–Se–Se#–Se–). A Se# atom can rotate dihedrally about the line joining its two neighboring Se without influencing the deformation pattern of the network (Figure 2). This is one example where certain internal degrees of freedom decouple from the rigidity of the overall network. In another example the extra degrees of freedom associated with a singly coordinated atom (i.e. a dangling vertex) decouple from the rigidity of the overall network. Clearly such decoupled degrees of freedom can be disregarded as far as network rigidity is concerned. The network, therefore, can satisfy the isostaticity condition in the range x < 0.2 by forming chemically ordered networks with an appropriate value of the k for the Se‐chains. A value of k = 3 corresponding to x = 1/7 (or about 0.14) is close to the composition where best glass formation is observed, explaining why good glass formation takes place for x < 0.15.
