# Theoretical Modelling of Ion Exchange Processes in Glass: Advances and Challenges
## 2 Basic Model
### Charge Neutrality Approximation in Field-Assisted Ion Exchange  
In the previous subsection, the charge neutrality was assumed in the field-assisted IE problem, as well as in the thermal-only case. However, this charge neutrality is not always fulfilled, especially when strong external electric fields are used and/or very high concentration gradients exist. In fact, some authors have modelled the silver concentration in channel waveguides by considering explicitly the space charge distribution [35], albeit at the cost of including other assumptions. An estimation of the validity of the charge neutrality approximation can be made from Equation (13), which can be expressed as:  
\[\frac{eE}{kT}=\
abla\ln(C_{A}+C_{B})+\frac{\frac{J_{0}}{(C_{A}+C_{B})D_{B}}- \
abla c_{A}}{1-\alpha c_{A}}\simeq\frac{\frac{I_{0}}{c_{0}D_{B}}-\
abla c_{A }}{1-\alpha c_{A}}, \tag{19}\]  
where we used the charge neutrality approximation and the definition of \(\alpha\). Now, we took divergences in this equation in order to compare it with Equation (10) and estimate the error made by this approximation. This is an iterative procedure. First, we used this approximated expression for the electric field. Next, we substituted it into [PERSON]'s Equation (10) to obtain a more accurate value for \(C_{A}+C_{B}\), which was introduced in the previous equation, and so on. However, a unique iteration will be enough to obtain an order of magnitude of the charge density [36]. Therefore, if we use Equation (12) and assume that \(\epsilon\) does not depend on \(C_{A}\) and \(C_{B}\), we obtained:  
\[\frac{e^{2}}{ekT}(C_{A}+C_{B})-\frac{e^{2}C_{0}}{ekT}\simeq\frac{\alpha J_{0} \
abla c_{A}}{(1-\alpha c_{A})^{2}C_{0}D_{B}}-\left(\frac{\alpha\
abla c_{A}} {1-\alpha c_{A}}\right)^{2}-\frac{\
abla^{2}c_{A}}{1-\alpha c_{A}}. \tag{20}\]  
Now, from this equation, we can obtain some conditions for the validity of the charge neutrality approximation by comparing each term on the right-hand side of this equation with the second term on the left-hand side. Therefore, for the second term on the right-hand side, this comparison provides:  
\[\frac{\alpha\
abla c_{A}}{1-\alpha c_{A}}\ll\sqrt{\frac{e^{2}C_{0}}{\epsilon_ {0}\epsilon_{r}kT}}\simeq 1.1\times 10^{10}\;\text{m}^{-1}, \tag{21}\]  
for a BK7 glass with a density of 2.4 g/cm\({}^{3}\) and 8.4% by weight of Na\({}_{2}\)O, which gives \(C_{0}\simeq 3.9\times 10^{27}\;\text{m}^{-3}\); likewise, typical values for the temperature, T = 400 \({}^{\circ}\)C, and for the relative permittivity, \(\epsilon_{r}=10\), were considered. As for the ion exchange, we chose for this assessment a Na\({}^{+}\)/K\({}^{+}\) IE in a soda-lime glass with a diffusion coefficient ratio \(D_{K}/D_{Na}\simeq 2.5\times 10^{-2}\)[37], which leads to \(\alpha=0.975\). Therefore, \(\
abla c_{A}\) must fulfil the following condition:  
\[\
abla c_{A}\ll 0.28\;\text{nm}^{-1} \tag{22}\]  
In the worst case, \(c_{A}=1\). This condition means a change, in the normalized concentration, from one to zero-point-nine at a distance much greater than 0.36 nm, which in practice is met in almost any IE process. Indeed, most IE processes use glasses with a higher Na\({}_{2}\)O content or their \(\alpha\) values are less than the one of Na\({}^{+}\)/K\({}^{+}\) IE. Therefore, this condition is even less restrictive in those cases. On the other hand, the third term on the right-hand side of Equation (20) is small under the same conditions as the second one.