## 3 Analysis of the Final Equilibrium State
### Calculation of Final Equilibrium State  
The intensity of the electric field inside the compact layer as well as the surface charge density \(\sigma\) near positive or negative electrode is calculated, according to the method of [PERSON] and [PERSON] [16]. Surface charge density (\(\varphi\)) is the amount of charge per unit of a two-dimensional surface area, and it is a measure of the quantity of electric charge accumulated over a surface. It is given by Equation (4):  
\[\sigma=-\sqrt{8\epsilon C_{M}RT}\text{sinh}[\frac{zF}{2 RT}(\frac{\sigma \lambda_{s}}{\varepsilon}+\varphi(0))] \tag{4}\]  
where \(C_{M}\) is the toxin concentration at the middle of the duct after the electric potential application, \(T\) is the absolute temperature which is considered \(T=300\) K throughout the study, \(z\) is the number of overflow protons or electrons (considered positive throughout the article), \(R=8.314\frac{\text{I}}{\text{mod}\text{K}}\), and \(F=96,485.34\) C/mol is the Faraday constant. We should at this point determine the \(C_{M}\) which is the toxin concentration at the middle of the duct after the electric potential application and it is therefore a critical value because it expresses the amount of residual toxin after the treatment. In this model, this parameter (\(C_{M}\)) was set at a specific value, according to the maximum levels of the toxins in water that can be tolerated by a human. Since there is much scientific literature regarding the intoxication by staphylococcal toxins, in this model, the minimum tolerable level was considered as reference value for staphylococcal poisoning. Children will suffer staphylococcal food poisoning by ingesting as little as 100 ng of staphylococcal enterotoxins, and only a few micrograms of staphylococcal enterotoxins are enough to cause poisoning in vulnerable populations [29]. Assuming that ingestion of 50 mL of water containing 100 ng of the staphylococcal enterotoxins could be harmful to a child, the level of 2 ng/mL was selected as the target concentration to be achieved in our model and this value is the  
Figure 4: Configuration of the double layer and equivalent capacitor representation, OHP: Outer Helmholtz Plane. \(C_{M}\) (accepted limits of the toxins). Moreover, this level can be relatively easily quantified with existing analytical methods such as ELISA [29; 30]. Taking into consideration the above reference value (2 ng/mL), the concentration of toxins \(C_{M}\) (considered approximately as the \"target\" concentration in the area of the duct away from the double layer after the treatment of the electric field), expressed in \(\
icefrac{{mol}}{{m^{3}}}\), was calculated as following:  
\[C_{M}=56\times 10^{-9\
icefrac{{mol}}{{m^{3}}}} \tag{5}\]  
The calculation of the Equation (5) value was performed using the values of the first toxin and for simplicity reasons was considered constant for all four toxins. Figure 5 presents the surface charge density (the discrete points and not the lines which are explained in Section 4) as a function of \(C_{M}\) for toxin 1 (_Staphylococcus aureus_ alpha-hemolysin) for various \(\varphi(0)\) (as we have already mentioned, \(\varphi(0)\) is the potential applied to one side of the duct). It is observed that the surface charge density remains stable by increasing \(C_{M}\). However, there is a large increase in surface charge density by increasing \(\varphi(0)\). The electric field intensity inside the compact layer is given by Equation (6). \[E_{\lambda_{S}}=-\frac{\sigma}{\varepsilon}\to E_{\lambda_{S}}=\sqrt{\frac{8C _{M}RT}{\varepsilon}}\text{sinh}[\frac{zF}{2 RT}(-\lambda_{s}E_{\lambda_{S}}+ \varphi(0))] \tag{6}\]  
but the same quantity outside the compact layer is described by Equation (7).