Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
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.
Figure 2 Schematic of a Se‐chain with five seleniums connecting two Ge atoms in an isostatic network (not shown). The inner Se# can rotate dihedrally about the dashed line connecting its neighboring seleniums without influencing the rigidity of the network. This dihedral motion of inner seleniums is an internal degree of freedom decoupled from the rigidity of the overall network.
For x > 0.2, there is experimental evidence for edge‐shared GeSe4 tetrahedra [29]. Because the presence of edge sharing does not change the value of r, the BCT results are not affected but edge sharing does cause differences in the intermediate‐range topology. When one considers a pair of edge‐shared tetrahedra as a single unit with four Se vertices (and two internal Ge atoms), such a bi‐tetrahedral unit has an internal degree of freedom, namely rotation about the shared edge. This additional flexibility allows isostaticity to hold beyond the GeSe2 composition (x = 1/3) in the PCT formalism even though the BCT results do not change. It is also clear that the presence of Ge–Ge homopolar bonds [16] – that must exist for x > 0.33 – does not influence the short‐range topology of the network. Hence, the BCT consequences do not change.
4.3.2 As–Se System
Since r(As) = 3 in the AsxSe(1−x) system, r* = 2.4 corresponding to x* = 0.4. But good glass formation has been observed in the Se‐rich range from x ~ 0 to x ~ 0.23. As in the Ge–Se system, this discrepancy can be rationalized by viewing the As–Se glasses for x < 0.4 as a chemically ordered network made up of linearly‐rigid (Sek) short chains, three of which being connected to every As atom. If one eliminates the internal degrees of freedom associated with the dihedral rotations of Se# in the Se chains, it follows that these chemically ordered As((Se)k)3/2 systems are isostatic for x ≤ 0.4. Note that since x = 2/(2 + 3 k), k ≥ 2 corresponds to x ≤ 0.25, which fits well the reported composition range for good glass formation [28].
4.4 Composition Variation of Properties in Glass‐forming Systems
Most properties of glasses exhibit rather uninteresting monotonic continuous variations even when r crosses its isostatic value. Only some configurational properties show extremum values with respect to r at the rigidity percolation threshold (i.e. at r* = 2.4). Tatsumisago et al. [30] reported that the configurational heat capacity and the activation energy of viscosity exhibited minima in the Ge–As–Se system at r = 2.4. Similar results were obtained by Senapati and Varshneya [31] in the Ge–Se and Ge–Sb–Se systems. It is worth noting that by investigating a range of Ge–As–Se compositions, all having the same values of r, Wang et al. [24] have reported that values of the configurational properties are not a unique function of r implying that topology alone is not sufficient to determine the variation of properties with composition, effect of chemical disorder must also be considered.
5 Temperature‐Dependent Constraints
5.1 The Influence of Thermal Energy
Implicit in the original PCT and BCT theories was the notion that constraints are fixed for good – either intact (= 1) or broken (= 0) – and that they do not vary with temperature (T). Thermal energy was implicitly neglected in the original theories which were thus valid only at T = 0 K. To remedy this problem, Gupta [5] introduced the concept of a T‐dependent bond constraint. He argued that, if Ei is the energy of a certain class of bonds, then the value of the corresponding constraint hi(T) should be expressed by a Boltzmann expression:
(12)
where kB is the Boltzmann constant. Note that the value of hi always lies in the interval [0,1], being zero in the high‐temperature limit, equal to 1 at sufficiently low temperatures, and decreasing monotonically with increasing T. Physically, a fractional value of a bond constraint means that only a fraction of ith type of bonds are intact at a given instant. One may associate a characteristic temperature Ti for the ith type of constraint as follows:
(13)