## 2 Materials and Methods  
### Mechanical Properties Calculations Using Makishima-Mackenzie Model  
The Makishima-Mackenzie model was applied to predict the elastic properties of the SiO\({}_{2}\)-Na\({}_{2}\)O-MgO-CaO-Al\({}_{2}\)O glasses doped with Sb\({}_{2}\)O\({}_{3}\) content. The present glass samples were collected from reference [5]. The composition, density, molecular weight, and molecular volume of the studied samples are presented in Table 1.  
The elastic moduli were theoretically predicted for the silicate glasses using [PERSON]'s model [20; 21]. The Makishima-Mackenzie model reports that [PERSON]'s modulus depends on the values of dissociation energy (\(G_{t}\)) and the packing density (\(V_{t}\)) of the metal oxides constituting the glass network:  
\[G_{t}\ \left(\text{kJ cm}^{-3}\right)=\sum_{i}X_{i}\ G_{i} \tag{1}\]  
where \(X_{i}\) and \(G_{i}\) refer to the molar fraction and the dissociation energy of the \(i\)th constituting compound. The \(G_{i}\) is related to the standard heat of formation (\(\Delta H_{j}\)) for each compound and determined from Equation (2).  
\[G_{i}\ \left(\text{kJ cm}^{-3}\right)=\frac{\rho_{i}}{M_{i}}\ \left(X\Delta H_{f}( metal)+Y\Delta H_{f}(oxygen)-\Delta H_{f}(compound)-(X+Y)RT\right) \tag{2}\]  
where \(\rho_{i}\), \(M_{i}\), \(\Delta H_{f}(metal)\), \(\Delta H_{f}(oxygen)\), \(\Delta H_{f}(compound)\), \(X\), and \(Y\) are the density of compound, molar weight of the compound, enthalpy of metal, enthalpy of oxygen, enthalpy of compound, the number of atoms of metal and oxygen, respectively, at the room temperature.  
\[V_{t}\left(\text{cm}^{3}\ \text{mol}^{-1}\right)=\frac{\rho}{M}\sum_{i}X_{i}\ V_{i} \tag{3}\]  
where \(V_{i}\) represents the packing factor of the \(i\)th constituent compound in the glass network.  
Based on the values of \(G_{t}\) and \(V_{t}\), the elastic properties (Young's modulus (\(Y\)), bulk modulus (\(B\)), shear modulus (\(S\)), longitudinal modulus (\(L\)), [PERSON]'s ratio (\(\mu\)), microhardness (\(H\)), and softening temperature (\(T_{g}\)) can be evaluated through the following formulas.  
\[Y=2V_{t}G \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Glass Code**} & \multicolumn{5}{c}{**Chemical Composition (wt\%)**} & \multicolumn{2}{c}{**Density**} & \multicolumn{2}{c}{**Molecular**} & \multirow{2}{*}{**Molar Volume**} \\ \cline{2-2} \cline{5-10}  & \(\text{Al}_{2}\text{O}_{3}\) & & & & & & & & \\ \hline SCBSb0 & 1.305 & 9.410 & 2.675 & 14.002 & 72.608 & 0.000 & 2.499 & 59.358 & 23.753 \\ SCBSb0.5 & 1.630 & 9.124 & 2.558 & 13.671 & 70.662 & 2.355 & 2.537 & 60.661 & 23.911 \\ SCBSb1 & 1.943 & 8.829 & 2.506 & 13.202 & 68.813 & 4.708 & 2.603 & 61.924 & 23.790 \\ SCBSb5 & 9.516 & 6.667 & 1.827 & 10.060 & 51.309 & 20.621 & 2.839 & 74.360 & 26.192 \\ SCBSb10 & 13.364 & 4.807 & 1.337 & 7.592 & 38.243 & 34.657 & 3.002 & 88.657 & 29.509 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The chemical composition, density molecular weight, and molar volume of the investigated glasses.  
\[B=1.2\ V_{t}E \tag{5}\] \[S=\frac{3\ YB}{(9B-Y)}\] (6) \[L=B+\frac{3}{4}\ S\] (7) \[\mu=\frac{Y}{2\ S}-1\] (8) \[H=\frac{(1-2\mu)}{6(1+\mu)}\] (9) \[T_{g}=\frac{M}{0.5074\ \rho\ v_{s}^{2}} \tag{10}\]