# Theoretical Modelling of Ion Exchange Processes in Glass: Advances and Challenges
## 3 The Mixed Ion Effect
### Generalized Equations and Boundary Conditions  
By following the same procedure as in the previous section, we replaced \(\mathbf{E}\) with \(\mathbf{J}_{0}\) in the expression of \(\mathbf{J}_{A}\), and then, we applied the continuity condition to obtain:  
\[\frac{\partial c_{A}}{\partial t}+\frac{\partial}{\partial c_{A}}\bigg{[} \frac{D_{A}c_{A}}{D_{A}c_{A}+D_{B}c_{B}}\bigg{]}\frac{D_{0}}{C_{0}}\
abla c_ {A}=\
abla\bigg{(}\frac{(g_{B}c_{A}+g_{A}c_{B})D_{A}D_{B}}{D_{A}c_{A}+D_{B}c_{ B}}\
abla c_{A}\bigg{)}, \tag{31}\]where the whole flux density \(\mathbf{J}_{0}\) can be obtained from the scalar potential \(\phi\) as:  
\[\mathbf{J}_{0}=-(D_{A}c_{A}+D_{B}c_{B})\frac{eC_{0}}{kT}\
abla\phi, \tag{32}\]  
and \(\phi\) in turn satisfies the following non-standard Laplace equation:  
\[\
abla[(D_{A}c_{A}+D_{B}c_{B})\
abla\phi]=0. \tag{33}\]  
In view of Equation (31), we can redefine the interdiffusion coefficient as:  
\[\bar{D}(c_{A})\equiv\frac{(g_{B}c_{A}+g_{A}c_{B})D_{A}D_{B}}{D_{A}c_{A}+D_{B}c _{B}}\equiv(g_{B}c_{A}+g_{A}c_{B})D_{\text{mob}}. \tag{34}\]  
It can be split into a mobility term \(D_{\text{mob}}\) and a thermodynamic term \((g_{B}c_{A}+g_{A}c_{B})\), which is not in the definition (9) of the basic model. If cations behave ideally, the latter becomes equal to one, and the interdiffusion coefficient is reduced to the mobility term. If cation interactions are relevant, the thermodynamic term enhances the maximum of \(\bar{D}(c_{A})\) for intermediate values of \(c_{A}\), as can be seen in Figure 1b. Similarly, Equation (32) shows that the mobility is:  
\[u\equiv\frac{e}{kT}(D_{A}c_{A}+D_{B}c_{B}), \tag{35}\]  
which, in fact, is the same as Equation (5), but now, the \(D_{i}\)'s are mole fraction dependent. This results in a minimum conductivity value for a mole fraction close to that at which the interdiffusion coefficient reaches its maximum. In contrast, the basic model leads to monotonic \(u\) and \(\bar{D}\) functions.  
In order to impose the boundary conditions on \(\phi\), we need to relate it with the electric potential \(v\). This can be done by the procedure followed in the above section, but the resulting expression is not so simple:  
\[\phi=V+\frac{kT}{e}\int_{0}^{c_{A}}\frac{D_{A}c_{A}\frac{\mathrm{d}\ln a_{A}}{ \mathrm{d}c_{A}}+D_{B}c_{B}\frac{\mathrm{d}\ln a_{B}}{\mathrm{d}c_{A}}}{D_{A} c_{A}+D_{B}c_{B}}\;\mathrm{d}c_{A}. \tag{36}\]  
If the MIE is caused by cation interactions (\(D_{i}c_{i}=D_{ii}a_{i}\)), this expression can be integrated for any particular dependency of the activities on the mole fraction:  
\[\phi=V+\frac{kT}{e}\ln\frac{D_{A}c_{A}+D_{B}c_{B}}{D_{BB}}.\]  
On the contrary, if cations behave ideally and relaxation processes are responsible for the MIE, then Equation (36) becomes:  
\[\phi=V+\frac{kT}{e}\int_{0}^{c_{A}}\frac{D_{A}-D_{B}}{D_{A}c_{A}+D_{B}c_{B}}\; \mathrm{d}c_{A},\]  
and the integration can only be performed once the dependencies of the diffusion coefficients on the mole fraction are known.  
The starting point to impose the boundary conditions are the same as in Section 2, that is equal electrochemical potentials at both sides of the glass surface for cation species that can cross it and zero flux through it otherwise. However, the particular functions for the electrochemical potentials and flux densities must be chosen from the model of the glass behaviour.