# Theoretical Modelling of Ion Exchange Processes in Glass: Advances and Challenges
## 3 The Mixed Ion Effect
### The Cation Flux Density  
Let us consider the electrochemical potential (\(\bar{n}_{i}\)) of each cation species that has a chemical term depending on the thermodynamic activity (\(a_{i}\)) of that species and an electric term proportional to the cation charge (\(e\)) and the electric potential (\(V\)):  
\[\bar{n}_{i}=\mu_{i}^{0}+kT\ln a_{i}+eV\qquad i=A,B, \tag{26}\]  
The flux density of each cation species is proportional to both the gradient of its electrochemical potential and its concentration. This leads to the Nernst-Planck equation:  
\[J_{i}=-D_{i}C_{i}\
abla\bar{n}_{i}=-g_{i}D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e \boldsymbol{E}}{kT}\qquad i=A,B, \tag{27}\]  
where the \(g_{i}\)'s are the thermodynamic factors:  
\[g_{i}\equiv\frac{\partial\ln a_{i}}{\partial\ln c_{i}}\qquad i=A,B.\]  
As mentioned above, some authors include the Haven ratio in the drift term of Equation (27). However, the Haven ratio should only be used to obtain \(D_{i}\) from the experimental values of the diffusion coefficient of radioactive tracers: \(D_{i}^{*}\)[74]. The difference between them arises, for example, if the diffusion mechanism is the indirect interstitial one. In this case, a cation located in one interstice replaces a nearby regular site cation, which jumps to another interstitial site. Consequently, the total mass (or charge) displacement described by \(D_{i}\) is different from that of a single cation, which is measured by the tracer. Moreover, the tracer is also affected by the thermodynamic factor, then a comparison between the mobility of a species and the corresponding tracer diffusion coefficient can result in an apparent Haven ratio [61]. If the MIE is fully due to the relaxations of the glass structure, the cation behaviour being ideal, the activity will be equal to the mole fraction, so \(g_{i}=1\). On the contrary, if the cation interaction is the only thing responsible for the MIE, we will obtain \(D_{i}=D_{ii}\gamma_{i}\), \(\gamma_{i}\) being the thermodynamic activity coefficient (\(a_{i}=\gamma_{i}c_{i}\)). Let us deduce that result. We divided all cations \(A\) into two sets, the ones that are hopping from one site to another in a given instant (\(A^{\uparrow}\)) and the rest of them, which are fixed (\(A^{\downarrow}\)). Although the former are much scarcer, both sets are in thermal equilibrium in any given small region in the glass; therefore:  
\[\bar{\mu}_{A}=\bar{\mu}_{A^{\downarrow}}=\bar{\mu}_{A^{\uparrow}} \tag{28}\]  
Now, we made two assumptions. First, we supposed that mobile cations behave ideally with respect to each other due their low concentration (\(C_{A^{\uparrow}}\ll C_{A^{\downarrow}}\)):  
\[\bar{\mu}_{A^{\uparrow}}=\mu_{A^{\uparrow}}^{0}+kT\ln\frac{C_{A^{\uparrow}}} {C_{0}}+eV\qquad i=A,B. \tag{29}\]  
Note that this could fail in the case of cooperative movement, that is, when two or more nearby cations change their site simultaneously. Second, we assumed that the reference potential of mobile cations (\(\mu_{A^{\uparrow}}^{0}\)) is independent of \(c_{A}\); therefore, \(\
abla\mu_{A^{\uparrow}}^{0}=0\). In addition to using Equation (27), the cation flux density can also be calculated from mobile cations, being proportional to both the gradient of their electrochemical potential and their concentration:  
\[\mathbf{J}_{A}=\mathbf{J}_{A^{\uparrow}}=-\frac{D_{A^{\uparrow}}}{kT}C_{A^{\uparrow}} \
abla\bar{\mu}_{A^{\uparrow}}=-D_{A^{\uparrow}}\
abla C_{A^{\uparrow}}+ \frac{eD_{A^{\uparrow}}}{kT}C_{A^{\uparrow}}E, \tag{30}\]  
where \(D_{A^{\uparrow}}\) is a temperature-dependent multiplicative coefficient and the factor \(1/kT\) was introduced in order for \(D_{A^{\uparrow}}\) to have units of a diffusion coefficient. By combining Equations (26), (28) and (29), we can find \(C_{A^{\uparrow}}\) as:  
\[C_{A^{\uparrow}}=\exp\left(-\frac{\mu_{A^{\uparrow}}^{0}-\mu_{A}^{0}}{kT} \right)\gamma_{A}C_{A},\]  
and replace it in Equation (30).