# Theoretical Modelling of Ion Exchange Processes in Glass: Advances and Challenges
## 2 Basic Model
### Nernst-Planck and Poisson Equations
The Haven ratio was ignored. (**a**) Basic model—Equations (5) and (9)—which assumes that the self-diffusion coefficients remain constant with the cation mole fraction. (**b**) The same functions taking into account the MIE. Quadratic polynomials in \(c_{K}\) were fitted to the logarithms of the experimental self-diffusion coefficients and then used to calculate the rest of the functions through the definitions (35) and (34). Note the difference between \(D\) and \(D_{\text{mob}}\), which are the expected interdiffusion coefficients when the interaction among cations or the ideal mixture is assumed, respectively. where \(\epsilon\) is the glass electrical permittivity and \(B\) is the magnetic field, which will be assumed as time independent since the total current changes very slowly. On the other hand, we also assumed the charge neutrality approximation (Equation (1)). This cannot be done in general, due to the existence of the aforementioned external field; however, in most cases, this is a very good approximation (see the next subsection). Doing this, the addition of continuity Equation (7) leads to:  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]1  
Now, by finding the electric field in Equation (4):  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]2  
taking into account Equation (12) and doing the same steps as before, the following electro-diffusion equation is obtained:  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]3  
which is an extension of Equation (8). On the other hand, Equation (13) can be inserted into (11), leading to:  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]4  
where we used Equation (9). This equation, Equation (12), and the boundary conditions, which will be presented later, determinate the flux density \(J_{0}\). From this flux density, Equation (14) will provide the evolution of the concentration of cations. Finally, once \(J_{0}\) and \(c_{A}\) are known, the charge neutrality approximation can be checked. This will be analysed in the following subsection. Alternatively, the last four equations can be expressed in terms of a scalar function. Indeed, an irrotational vector field is the gradient of a potential function \(\phi\), so:  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]5  
where the factor \(-e/(kT)\) was included in order for \(\phi\) to have the same units as the electric potential. From this equation and Equation (12), we obtained a non-standard Laplace equation:  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]6  
which is more convenient for resolution purposes than vector Equations (12) and (15). Note that this effective potential \(\phi\) is not the electric potential \(V\), whose minus gradient is the electric field \(E\) given by Equation (13). However, a relationship between them can be obtained [30] by inserting Equation (16) into Equation (13), that is:  
\[\mathbf{J}_{i}=-D_{i}\
abla C_{i}+D_{i}C_{i}\frac{e\mathbf{E}}{kT}\qquad i=A,B\;; \tag{3}\]7  
Under typical IE conditions, the difference between \(V\) and \(\phi\) is no greater than a few tenths of a volt, which is negligible compared to the usual voltages (20-100 V) used in field-assisted IE. Despite this, this difference must not be ignored in the modelling, as significant errors in the calculation of the electric field could be made. Indeed, although both potentials are similar, their gradient is not always. On the other hand, it is worth mentioning that Equations (12) and (15) are trivially solved in the one-dimensional case, that is \(J_{0}\) is constant. This constant will be established from the experimental setup. If a constant current source is used, this value is set directly.