## 4 Theoretical  
Let us expand the temperature-dependent activation energy of viscous flow \(E(T)\) near the \(T_{g}\) into a Taylor series in terms of a small temperature increment with a small parameter available in the system \(\lambda=(T-T_{g})/T_{g}<<1\), limiting the expansion to the first three terms of the series [24]: \begin{table}
\begin{tabular}{c c c c c c c} \hline \multirow{2}{*}{**N**} & \multicolumn{3}{c}{**Composition, mol.\%**} & \multirow{2}{*}{\(T_{g}\), **K**} & \multirow{2}{*}{\(C_{1}\)} & \multirow{2}{*}{\(C_{2}\), **K**} & \multirow{2}{*}{\(E_{g}\), kJ/mol**} \\ \cline{2-2} \cline{5-8}  & **As** & & & & & \\ \hline  
\hline \end{tabular}
\end{table} 
Table 1: Parameters of WLF Equation (4) and activation energy of the glass transition process at \(T=T_{g}\) (data from Refs. [20; 21; 22; 23; 24; 25] were used). Figure 4: Temperature dependence of the activation energy of viscous flow \(E(T)\) for Ge-Se molten glass. Ge content—92 mol.%, Se—8 mol.%. Points and experimental data from [22], curve calculation using Formula (9), \(T_{g}=352\) K.  
\[E=E_{g}-D_{1}\big{(}T-T_{g}\big{)}+D_{2}(T-T_{g})^{2}\text{ where }D_{1}=-\bigg{(}\frac{\partial E}{\partial T}\bigg{)}_{T_{g}},\ D_{2}=\frac{1}{2} \bigg{(}\frac{\partial^{2}E}{\partial T^{2}}\bigg{)}_{T_{g}}. \tag{12}\]  
The small parameter in expansion (12) is the dimensionless quantity \(\lambda\), but \(T_{g}\) to the first and second powers is hidden in the coefficients \(D_{1}\) and \(D_{2}\). From Equation (1), it follows that:  
\[-\frac{\partial ln\ \eta}{\partial\big{(}T/T_{g}\big{)}}=-\frac{T_{g}}{R}\bigg{[} \frac{1}{T}\bigg{(}\frac{\partial E}{\partial T}\bigg{)}-\frac{E}{T^{2}}\bigg{]} \tag{13}\]  
According to Equation (5), for parameter \(A\), we obtain the relation:  
\[A=\left.-\frac{\partial ln\ \eta}{\partial\big{(}T/T_{g}\big{)}}\right|_{T=T_{g} }=-\frac{1}{R}\bigg{[}\bigg{(}\frac{\partial E}{\partial T}\bigg{)}_{T_{g}}- \frac{E_{g}}{T_{g}}\bigg{]}=\frac{D_{1}}{R}+\frac{E_{g}}{RT_{g}}, \tag{14}\]  
where we accounted for the definition of \(D_{1}\) from (12). The parameter \(A\) is given in Equations (5) and (6) by:  
\[A=\frac{C_{1}}{C_{2}}T_{g} \tag{15}\]  
Equations (14) and (15) in turn give the following equality:  
\[D_{1}=\frac{C_{1}RT_{g}}{C_{2}}-\frac{E_{g}}{T_{g}}, \tag{16}\]  
which considering \(E_{g}=C_{1}RT_{g}\) transforms into the following equation:  
\[D_{1}=C_{1}R\bigg{(}\frac{T_{g}}{C_{2}}-1\bigg{)} \tag{17}\]  
We find the formula for coefficient \(D_{2}\) from the calculations as follows:  
\[\frac{1}{2}\frac{\partial^{2}ln\ \eta}{\partial\big{(}T/T_{g}\big{)}^{2}}=\frac{T_{g} ^{2}}{2}\frac{\partial}{\partial T}\bigg{(}\frac{\partial ln\ \eta}{\partial T}\bigg{)}=\frac{T_{g}^{2}}{2R}\frac{ \partial}{\partial T}\bigg{[}\frac{1}{T}\frac{\partial E}{\partial T}-\frac {E}{T^{2}}\bigg{]}=\frac{T_{g}^{2}}{2R}\bigg{\{}\frac{1}{T}\bigg{(}\frac{ \partial^{2}E}{\partial T}\bigg{)}-\frac{1}{T^{2}}\bigg{(}\frac{\partial E}{ \partial T}\bigg{)}-\frac{1}{T^{2}}\bigg{(}\frac{\partial E}{\partial T} \bigg{)}+\frac{2E}{T^{3}}\bigg{\}} \tag{18}\]  
The coefficient \(B\) is given accordingly by:  
\[B=\frac{1}{2}\frac{\partial^{2}ln\ \eta}{\partial\big{(}T/T_{g}\big{)}^{2}} \bigg{|}_{T=T_{g}}=\frac{T_{g}}{R}\bigg{\{}D_{2}+\frac{D_{1}}{T_{g}}+\frac{E _{g}}{T_{g}^{2}}\bigg{\}} \tag{19}\]  
Further, in this relation for \(D_{1}\), we consider expression (16), resulting in:  
Figure 4:0  
From (6) and (15), we have equality \(B=C_{1}(T_{g}/C_{2})^{2}\), which, on substitution into (20), finally gives \(D_{2}\):  
Figure 4:1  
If we go a slightly different route and use a joint transformation of expressions (12) and (1) with the concomitant finding of differentials \(A\) and \(B\), then we can obtain equivalent data. Thus, the activation energy of viscous flow is represented in the form of a Taylor series in the temperature range (2) in the form:\[