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)
so that this constraint can be considered effectively as broken (= 0) for T > Ti and intact (=1) for T < Ti. For both stretching and bending constraints, this formalism has been validated by comparisons of the standard deviations of the partial distributions calculated for glassy and crystalline alkali disilicates as a function of temperature in MD simulations [9].
The average degree of freedom per vertex, f(T), in the network thus becomes T‐dependent and (for d = 3) is given by
(14)
Since hi is a decreasing function of T, f(T) always increases with temperature.
5.2 Extension of the Topological Constraint Theory to Supercooled Liquids
In 1999, Gupta [6] extended the notion of T‐dependent bond constraints to glass‐forming supercooled liquids: “Since the structure of a glass formed by cooling a liquid is the same as the structure of the liquid at the glass transition (or fictive) temperature, Tg , it follows that if the glass structure is an extended TD network, then such a network must also exist in the super‐cooled liquid state at Tg .” More importantly, he argued that the configurational entropy, ΔS(T), of a supercooled liquid is approximately proportional to the average degrees of freedom per vertex, f(T). This result, later substantiated by Naumis [32], leads to several important consequences:
1 At the Kauzmann temperature, TK, defined by ΔS(TK) = 0, the degrees of freedom vanish:(15)
1 From the Adam–Gibbs theory of viscosity, it follows that the temperature‐dependence of viscosity is simply related to that of f(T):(16)
Here, A is a constant independent of T.
1 The fragility, m, of a liquid defined as(17)
is related to the temperature‐dependence of f as follows:
(18)
The value of log [ηg/η∞] is about 16. The variation of the degrees of freedom, f(T), with T, for good, poor, and non‐glass‐forming liquids is shown schematically in Figure 3. From Eq. (16), one then concludes that the cause of the non‐Arrhenian
