# Theoretical Modelling of Ion Exchange Processes in Glass: Advances and Challenges
## 2 Basic Model
### Some Particular Solutions  
Two noticeable one-dimensional solutions are obtained for both specific thermal and field-assisted IE processes. In the simplest thermal IE, a glass sheet is immersed in a molten salt, which contains foreign cations. The boundary conditions for \(c_{A}\) and \(\phi\) are constant over time and along the glass surface, then \(J_{0}=0\). The concentration profile of the dopants scales in depth proportionally to the square root of the diffusion time, but it retains its shape, that is \(c(x,t)=c_{A}|_{S}f(x/\sqrt{4D_{A}t})\), being the origin of the coordinates at the glass surface [44]. The shape of \(f\) depends on \(c_{A}|_{S}\) and \(\alpha\). If either of them is small, the diffusion Equation (14) becomes linear and \(f\) tends to a complementary error function (erfc), which has an inflection point at the glass surface and decreases monotonically to zero inside the glass. Otherwise, the diffusion is faster near the surface, where the dopant concentration is high. This causes \(f\) to present a bump near the surface, being the inflection point displaced into the glass. \(f\) is often approximated by a Gaussian function, although other functions have been proposed [45; 46; 47]. \begin{table}
\begin{tabular}{c c c} \hline  & \(c_{A}\) & \(J_{0}\) \\ \hline Equations & \(\frac{\partial c_{A}}{\partial t}+\frac{I_{0}}{C_{0}}\frac{1-\alpha}{(1- \alpha c_{A})^{2}}\
abla c_{A}=\
abla(\tilde{D}(c_{A})\
abla c_{A})\) & \(\left\{\begin{array}{l}\
abla J_{0}=0\\ \
abla\times\left(\frac{J_{0}}{D_{B}C_{0}(1-\alpha c_{A})}\right)=\mathbf{0} \end{array}\right.\) \\ Salt & \(c_{A}|_{S}=\mathrm{C}\) & \(\left\{\begin{array}{l}\mathbf{J_{0}}\times\hat{\varepsilon}_{S}=\mathbf{0} \\ \int_{I_{0}}\mathbf{d}\mathrm{S}=I_{i}\end{array}\right.\) \\ Film & \(\
abla c_{A}|_{S}\cdot\hat{\varepsilon}_{S}=\frac{(c_{A}|_{S}-1)J_{0}|_{S} \cdot\hat{\varepsilon}_{S}}{D_{A}C_{0}}\) & \(\left\{\begin{array}{l}\mathbf{J_{0}}\times\hat{\varepsilon}_{S}=\mathbf{C_ {0}}D_{B}\frac{\
abla c_{A}\times\hat{\varepsilon}_{S}}{c_{A}}\\ \int_{I_{i}}J_{0}\mathrm{d}\mathrm{S}=I_{i}\end{array}\right.\) \\ Mask/Air & \(\
abla c_{A}|_{S}\cdot\hat{\varepsilon}_{S}=0\) & \(\mathbf{J_{0}}\cdot\hat{\varepsilon}_{S}=0\) \\ \hline with & \(\sum_{i}I_{i}=0\) & \\ \hline \end{tabular}
\end{table} 
Table 1: Complete formulation of the IE problem (equations and boundary conditions), for the most common IE processes, in terms of the normalized concentration of dopant cations \(c_{A}\) and the total flux density \(J_{0}\); \(\hat{\varepsilon}_{S}\) is a unit vector perpendicular to the sample surface, and \(\mathrm{C}\) is a constant, which mainly depends on the dopant concentration in the salt. \begin{table}
\begin{tabular}{c c c} \hline  & \(c_{A}\) & \(\phi\) \\ \hline Equations & \(\frac{\partial c_{A}}{\partial t}+\frac{I_{0}}{C_{0}}\frac{1-\alpha}{(1- \alpha c_{A})^{2}}\
abla c_{A}=\
abla(\tilde{D}(c_{A})\
abla c_{A})\) & \(\
abla((1-\alpha c_{A})\
abla\phi)=0\) \\ Salt & \(c_{A}|_{S}=\mathrm{C}\) & \(\phi|_{S}=\mathrm{F}\) \\ Film & \(\
abla c_{A}|_{S}\cdot\hat{\varepsilon}_{S}=\frac{(c_{A}|_{S}-1)J_{0}|_{S} \cdot\hat{\varepsilon}_{S}}{D_{A}C_{0}}\) & \(\phi|_{S}=\mathrm{F}+\frac{kT}{e}\ln\Bigl{(}\frac{1}{c_{A}|_{S}}-\alpha\Bigr{)}\) \\ Mask/Air & \(\
abla c_{A}|_{S}\cdot\hat{\varepsilon}_{S}=0\) & \(\
abla\phi|_{S}\cdot\hat{\varepsilon}_{S}=0\) \\ \hline with & \(\frac{J_{0}}{D_{B}C_{0}}=-\frac{e}{kT}(1-\alpha c_{A})\
abla\phi\) \\ \hline \end{tabular}
\end{table} 
Table 2: Complete formulation of the IE problem (equations and boundary conditions), for the most common IE processes, in terms of the normalized concentration of dopant cations \(c_{A}\) and an effective electric potential \(\phi\); \(\hat{\varepsilon}_{S}\) is a unit vector perpendicular to the sample surface, and \(\mathrm{C}\) and \(\mathrm{F}\) are constants. When a voltage is applied between both surfaces of the glass sheet, a current normal to these surfaces appears.