## 3 The Temperature Interval of the Liquid-Glass Transition  
If we rewrite the WLF Equation (3) as a straight line equation and draw graphs in the coordinates \(\mathbf{y}=-[(T-T_{g})/\ln(a_{T})]\) and \(\mathbf{x}=(T-T_{g})\); then, we can get a linear dependence of \(\mathbf{y}\) on \(\mathbf{x}\). Figure 1 shows that the dependence \(\mathbf{y}(\mathbf{x})\) in the glass transition region is linear, which confirms the applicability of the WLF equation. Experimental data on the temperature dependence of the viscosity of chalcogenide glasses were used from the electronic database SciGlass [18], which are shown in Table 2.  
Similarly, graphs were constructed for all compositions of studied polymers. From these straight lines \(\mathbf{y}\)-\(\mathbf{x}\), the values of the parameters of the WLF equation \(C_{1}\) and \(C_{2}\) are determined (Table 1). Within [PERSON]'s model of delocalized atoms [8; 17; 28], an interpretation of the WLF equation is proposed, from which it follows that the parameter \(C_{1}\) is the reciprocal of the fraction of the fluctuation volume \(f_{g}\) frozen at the glass transition temperature:  
\[C_{1}=\frac{1}{f_{g}} \tag{5}\]  
Figure 1: Temperature dependence of the viscosity of As-S glasses in coordinates corresponding to the Williams–Landel–[PERSON] equation. The data of [18] were used where the composition is as follows: As—32.5 mol%; 5—67.5 mol%. The fluctuation volume of an amorphous substance \(\Delta\)V\({}_{e}\) is due to thermal displacements of atoms (kinetic units) [17; 28]  
\[\Delta\)V\({}_{e}=N_{e}\Delta\)v\({}_{e}, \tag{6}\]  
where \(N_{e}\) is the number of delocalized atoms, and \(\Delta\)v\({}_{e}\) is the elementary fluctuation volume required for the delocalization of an atom, i.e., its maximum displacement from the local equilibrium position. The volume fraction of the fluctuation volume \(f_{S}\), obtained from the data on the parameter C\({}_{1}\), of course, weakly depends on the nature of amorphous substances [8; 17] (Table 1) and is no higher than 10%. \[f_{S}=\left(\frac{\Delta\)V\({}_{e}}{\)V\({}_{T=T_{S}}}\approx const\approx 0.03-0.08\right. \tag{7}\]  
During cooling in the glass transition region, due to the slowing down of molecular rearrangement, the change in the structure of the melt does not have time to follow the change in temperature. In the glass transition region, the viscosity of the melt sharply increases, as well as the relaxation time \(\tau(T)\), which, in turn, is related to the liquid cooling rate \(q=(dT/dt)\). According to relaxation theories [5; 29; 30; 31; 32; 33; 34], the relationship between the relaxation time and the melt cooling rate plays a decisive role in the glass transition process. [PERSON] [29], in 1951, from general considerations based on experimental data, proposed an equation relating these quantities in the following form:  
\[q\tau_{S}=C, \tag{8}\]  
where \(\tau_{S}\) is the structural relaxation time at the glass transition temperature \(T_{S}\), and \(C\) is an empirical parameter with the dimension of temperature. A rigorous physical theory about the behavior of kinetic units that can be in two states with different states separated by an energy barrier was developed by [PERSON] and [PERSON] [30]. The solution of the kinetic equation shows that the glass transition of the system occurs at a certain temperature \(T_{S}\) when the fraction of particles in the excited state freezes. The temperature \(T_{S}\) satisfies the condition:  
\[\left(\frac{d\tau}{dT}\right)_{T=T_{S}}=-\frac{1}{q} \tag{9}\]  
The kinetic criterion for the liquid-glass transition in this theory is the glass transition equation [28]:  
\[q\tau_{S}=\delta T_{S} \tag{10}\]  
The value of \(\delta T_{S}\), which was proposed in this notation by [PERSON] [31], characterizes the temperature range of the transition from liquid to glass upon cooling. To estimate this parameter of the glass transition equation \(\delta T_{S}\) for the glasses and polymers under study, we turn to expressions (3) and (9) discussed earlier.