# Theoretical Modelling of Ion Exchange Processes in Glass: Advances and Challenges
## 2 Basic Model
### Nernst-Planck and Poisson Equations  
The initial non-homogeneity of cation concentrations, \(C_{A}(x,t)\) and \(C_{B}(x,t)\), and their random motion make them diffuse along the glass until their concentrations are homogeneous. This diffusion process is described by [PERSON]'s law:  
\[\mathbf{J}_{i}^{\text{\emph{dif}}}=-D_{i}\
abla C_{i}\qquad i=A,B\;; \tag{2}\]  
where \(\mathbf{J}_{i}\) is the flux density and \(D_{i}\) the diffusion coefficient of each cation. However, the two interdiffusing cations have usually different diffusion coefficients and, therefore, different mobilities, which produce charge imbalances. These imbalances generate a strong internal electric field (\(\mathbf{E}\)), which tends to balance the charges and restore charge neutrality (Equation (1)). Therefore, [PERSON]'s equation is no longer valid, and a drift term must be added on its right-hand side to take into account the effect of this electric field on the cation motion. This leads to the Nernst-Planck equation [34]:  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]  
where \(e\) is the proton charge, \(T\) the absolute temperature and \(k\) [PERSON]'s constant. Note that some authors included, in the drift term of this equation, the Haven ratio. However, as we will see below, this parameter should not be incorporated into the model as a general rule. The occurrence of the above-mentioned electric field can be seen from a quantitative point of view by calculating the total flux density:  
\[\mathbf{J}_{0}=\mathbf{J}_{A}+\mathbf{J}_{B}=-D_{A}\
abla C_{A}-D_{B}\
abla C_{B}+(D_{A}C_ {A}+D_{B}C_{B})\frac{e\mathbf{E}}{kT} \tag{4}\]  
which depends on the electric field through mobility \(u\):  
\[u=\frac{e}{kTC_{0}}(D_{A}C_{A}+D_{B}C_{B}). \tag{5}\]  
However, \(\mathbf{J}_{0}\) cancels in the current problem, that is,  
\[\mathbf{J}_{0}=\mathbf{0}, \tag{6}\]  
because there is no external field applied that generates a net current. This means that the imbalance of the diffusion of the two cations (\(D_{A}\
abla C_{A}+D_{B}\
abla C_{B}\)) is compensated by the internal field through the drift term \(uC_{0}\mathbf{E}\). Note that Equation (4) is a generalization of [PERSON]'s law. On the other hand, as long as there is no creation or destruction of cations from/to a metallic state, the continuity equation must be fulfilled:  
\[\frac{\partial C_{i}}{\partial t}+\
abla\mathbf{J}_{i}=0\qquad i=A,B. \tag{7}\]  
Now, by combining this equation and Equation (3) for, for instance, \(i\)=A and substituting, in the resulting equation, the electric field from Equations (4) and (6), we obtained the differential equation that gives the concentration evolution [23] of cation A:  
\[\frac{\partial c_{A}}{\partial t}=\
abla(\bar{D}(c_{A})\
abla c_{A}), \tag{8}\]  
where the charge neutrality (Equation (1)) was applied and a normalized concentration (\(c_{A}=C_{A}/C_{0}\)) was used, and we defined the following interdiffusion coefficient:  
\[\bar{D}(c_{A})=\frac{D_{A}}{1-\alpha c_{A}}, \tag{9}\]where \(\alpha=1-D_{A}/D_{B}\). The dependence of \(u\) and \(\bar{D}\) on \(c_{K}\) predicted by this basic model, for K\({}^{+}/\)Na\({}^{+}\) IE, can be seen in Figure 1a. A more complex problem is the IE assisted by an external electric field. In such a case, Equation (6) is not met. Therefore, additional equations are necessary to calculate the total flux density (\(J_{0}\)) or, alternatively, \(\mathbf{E}\), which now includes the external field. As for \(\mathbf{E}\), it must fulfil [PERSON]'s equation and [PERSON]'s law of induction:  
\[\
abla(\epsilon\mathbf{E})=e(C_{A}+C_{B}-C_{0}) \tag{10}\] \[\
abla\times\mathbf{E}+\frac{\partial\mathbf{B}}{\partial t}=\mathbf{0}\, \tag{11}\]  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]0 Self-diffusion and interdiffusion coefficients, as well as mobility, calculated from experimental data obtained by radiative tracers [27], as a function of the normalized concentration for K\({}^{+}/\)Na\({}^{+}\) IE.