## 2 Theoretical Background: The Lever Rule and the Short-Range Order Configuration Model
### The Conventional Lever Rule  
Since pure B\({}_{2}\)O\({}_{3}\) glass consists entirely of neutral borate triangles (B\(\varnothing_{3}\)), the progressive addition of a modifying metal oxide will result in the gradual transformation of neutral triangles initially into meta-type units, namely B\(\varnothing_{2}\)O\({}^{-}\) or B\(\varnothing_{4}\)\({}^{-}\); then into pyroborate triangles, namely \(\text{B}\text{O}\text{O}_{2}{}^{2-}\); and finally into ortho-type moieties, namely \(\text{BO}\text{O}_{3}{}^{3-}\) or \(\text{B}\text{O}_{2}\text{O}_{2}{}^{3-}\). The peculiarity of the boron atom to change its coordination number from 3 to 4 results in a slight modification of the conventional lever rule. In that sense, upon initial introduction of a modifier in the fully polymerized \(\text{B}_{2}\text{O}_{3}\) glass, it is possible to create the metaborate tetrahedra \(\text{B}\text{O}_{4}{}^{-}\), where all oxygen atoms are bridging, and thus, non-bridging oxygens are absent. The previous gradual structural transformations dictate that in the composition range of a binary glass with the chemical formula xMO-(1-x)\(\text{B}_{2}\text{O}_{3}\), where M denotes a divalent cation, four **fundamental** nodes must be considered. These include the neutral, N (x = 0); the meta, M (x = 0.50); the pyro, P (x = 0.67); and the ortho, O (x = 0.75), nodes. Such nodes result in the partition of the composition range in three intervals, i.e., 0 \(\leq\) x \(\leq\) 0.50, 0.50 \(\leq\) x \(\leq\) 0.67, and 0.67 \(\leq\) x \(\leq\) 0.75. According to the lever rule, in each of these intervals, only two types of units can be present, specifically the ones representing the limits of the corresponding interval. For example, in the 0.50 \(\leq\) x \(\leq\) 0.67 range, between M and P nodes, the lever rule predicts that the network structure should consist only of metaborate and pyroborate entities. It is noted that the previous remarks hold also in the case of xM\(\text{}_{2}\)O-(1-x)\(\text{B}_{2}\text{O}_{3}\) glasses, where M is a monovalent metal ion. At this point, we should state the first two prerequisites and necessary conditions that emerge from the lever rule:  
**Condition I**: In the composition range, four nodes are fundamental. These correspond to the neutral, N; meta, M; pyro, P; and ortho, O, nominal compositions. **Condition II**: At each interval between the fundamental nodes, only two types of structural units are allowed. These units correspond to the two fundamental nodes that define the interval. Once the previous conditions are established, it is straightforward to calculate the corresponding molar fractions of the various borate building units by invoking only mass and charge balance considerations. This is feasible due to condition II since there are two equations with two unknown molar fractions. It is mentioned in the original paper of the SROC model [44] that instead of the charge conservation, the equivalent representation of the ratio of oxygen atoms to boron, namely \(A(x)\), was adopted. Denoting by \(X_{N},\ X_{M},X_{P}\), and \(X_{O}\) the molar fractions for neutral, meta, pyro, and orthoborate species, respectively, the following equations are easily derived (for analytical details, see [44]):  
**Range I** (0 \(\leq x\leq\) 0.50)  
\[X_{N}(x)=\frac{1-2x}{1-x} \tag{3a}\] \[X_{M}(x)=\frac{x}{1-x} \tag{3b}\]
**Range II** (0.50 \(\leq x\leq\) 0.67)  
\[X_{M}(x)=\frac{2-3x}{1-x} \tag{4a}\] \[X_{P}(x)=\frac{2x-1}{1-x} \tag{4b}\]
**Range III** (0.67 \(\leq x\leq\) 0.75)  
\[X_{P}(x)=\frac{3-4x}{1-x} \tag{5a}\] \[X_{O}(x)=\frac{3x-2}{1-x} \tag{5b}\]  
The functions \(X_{N}(x),\ X_{M}(x),X_{P}(x)\), and \(X_{O}(x)\) are depicted in Figure 1.  
At this point, it is also evident that each function of the molar fractions assumes a maximum value of 1 at the corresponding node, i.e., \(X_{M}(x)=1\) at \(x=0.5\) (M node), and in addition, it reaches the value 0 at the adjacent nodes, i.e., \(X_{M}(x)=0\) at \(x=0\) (N node) and \(x=0.67\) (P node).