Encyclopedia of Glass Science, Technology, History, and Culture. Группа авторов

Figure 3 Schematic variation of degrees of freedom (f) in three supercooled liquids with increasing temperature normalized with respect to the Kauzmann temperature (T_K). 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, T_g

      If the value of the parameter A in Eq. (16) has a negligible composition dependence, then it follows from this equation that

(21)

Here, x_ref is a reference composition. The importance of Eq. (21) cannot be overstated. It provides a means of modeling the composition dependence of T_g 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, T_g(x), with composition in the binary Ge_x 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 T_g 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 analyzed the four component Na₂O–CaO–B₂O₃–SiO₂ system [10].

Figure 4 Composition dependence of the glass transition temperature for the (a) sodium borate and (b) lithium borate systems. Solid curves: predicted T_g(x) using the temperature‐dependent TCT. Points: experimental data

      (Source: From [8]).