Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов
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 nature of the viscosity is the temperature‐dependence of bond constraints.
Figure 3 Schematic variation of degrees of freedom (f) in three supercooled liquids with increasing temperature normalized with respect to the Kauzmann temperature (TK). Curve (a) represents a strong glass former, curve (b) a fragile glass former, and curve (c) a non‐glass former for which a TD network cannot exist
(Source: From [6]).
5.3 Temperature – Scaling of Viscosity (η) and the MYEGA Equation
Substituting Eqs. (12) and (14) in Eq. 16 and assuming that only one type of constraints (with n constraints per vertex) varies within the temperature range of interest, one obtains the following temperature scaling of viscosity for supercooled liquids:
(19)
For deeply supercooled liquids in the vicinity of the glass transition, n is approximately equal to 3 and Eq. (16) simplifies to
(20)
where B is a new constant. This Eq. (20) is the well‐known MauroYue‐Ellison‐Gupta‐Allan (MYEGA) expression [33] for the T‐dependence of the equilibrium viscosity, which has been remarkably successful in fitting the experimental data on the T‐dependence of viscosity for a large number of inorganic and organic liquids. It should be noted that, like the Vogel‐Tammann‐Fulcher (VFT) equation, the MYEGA has only three fitting parameters but, unlike VFT, it does not exhibit any divergence of viscosity at any finite temperature.
5.4 The Composition Variation of the Glass Transition Temperature, Tg
If the value of the parameter A in Eq. (16) has a negligible composition dependence, then it follows from this equation that
(21)
Here, xref is a reference composition. The importance of Eq. (21) cannot be overstated. It provides a means of modeling the composition dependence of Tg from the knowledge of the atomic level short‐range order as a function of composition. Traditionally, such information on the short‐range order has been obtained from X‐ray or Neutron diffraction studies. Nowadays, such information can also be obtained accurately using MD studies.
Gupta and Mauro [7] used the T‐dependent constraint theory to rationalize quantitatively the variation of the glass transition temperature, Tg(x), with composition in the binary Gex Se(1−x) chalcogenide system. Their analysis resulted in the modified Gibbs–DiMarzio equation:
(22)
with a value of the parameter α = 5/3, which is the value observed experimentally [31]. In addition, using Eq. (18), Gupta and Mauro [7] were also able to explain the variation of the fragility, m, as a function of composition.
Mauro et al. [8] later applied the T‐dependent constraint theory to binary alkali–borate systems where a wealth of structural information is available as a result of years of X‐ray diffraction and NMR spectroscopy experiments. The agreement for both Tg and m between TCT and experimental results is remarkable considering that only one fitting parameter was used for all the data (see Figures 4 and 5). Using a similar approach, Smedskjaer et al. [34] have successfully extended the T‐constraint theory to ternary system Na2O–CaO–B2O3 system. In 2011, they also