## 3 Analysis of the Final Equilibrium State
### Calculation of Final Equilibrium State
\[E=-\frac{\partial\varphi}{\partial y}=\sqrt{\frac{8C_{M}RT}{\varepsilon}} \text{sinh}\left(\frac{zF}{2 RT}\varphi\right) \tag{7}\]  
where  
\[\varphi=\frac{4 RT}{zF}tanh^{-1}\left\{tanh(\frac{zF\varphi_{s}}{4 RT})\cdot e^{ -\kappa(y-\lambda_{s})}\right\}\text{ for }\rightarrow\lambda_{s}\leq y\leq\
icefrac{{ L}}{{2}} \tag{8}\]  
Figure 5: Surface charge density \(|\sigma|\left(\text{C/m}^{2}\right)\) as a function of \(C_{M}\left(\
icefrac{{mol}}{{m^{3}}}\right)\) for toxin 1 (_Staphylococcus aureus_ alpha-hemolysin) for various electric potentials \(\varphi(0)\) (\(V\)) (discrete points). Solid lines represent the corresponding \(|\sigma_{1}|\) values. \[\varphi=\frac{4 RT}{zF}tanh^{-1}\bigg{\{}-tanh(\frac{zF\varphi_{s}}{4 RT})\cdot e^{x(y -L+\lambda_{s})}\bigg{\}}\text{ for }\rightarrow\
icefrac{{L}}{{2}}\leq y\leq L-\lambda_{s} \tag{9}\]  
where \(\lambda_{s}\) is replaced on a case-by-case basis (Table 1) and the potential \(\varphi_{s}\) at the outer Helmholtz plane (OHP) is related with \(\varphi(0)\) with Equation (10). \[\varphi_{S}=\varphi(0)-\lambda_{s}\sqrt{\frac{8C_{M}RT}{\varepsilon}}\text{ sinh}(\frac{zF}{2 RT}\varphi_{s}) \tag{10}\]  
In order to determine the area that is occupied by the excess of the charged toxins that were moved, and consequently the area of the liquid whose concentration is within acceptable limits, we must calculate the total differential capacitance \(c_{tot}\), defined as  
\[c_{tot}=\frac{|d\sigma|}{d\varphi(0)}\]  
Using Equation (4), we have:  
Figure 4:0  
According to what has already been mentioned, the boundary layer consists of two capacitors connected in-line, the first corresponding to the compact layer (Stern layer) with capacity \(c_{H}\) and the second that to the diffuse layer with capacity \(c_{D}\)  
Figure 4:1  
Figure 4:2  
Moreover, \(c_{D}\) can be alternatively expressed as:  
Figure 4:3  
where  
Figure 4:4  
is the diffuse layer width in the linear regime. It is evident from Equation (12) that \(c_{D}\) depends on the initial concentration and the applied electric field. By using Equation (13), \(\lambda_{D}\) values for all toxins were calculated and are presented in Figure 4:5 \begin{table}
\begin{tabular}{c c c c c} \hline \hline \begin{tabular}{c} **Toxin 1** \\ _Staphylococcus aureus_, \\ **Alpha-Hemolysin** \\ \end{tabular} & \begin{tabular}{c} **Toxin 2** \\ _Staphylococcus aureus_, \\ **Toxic Shock Syndrome** \\ **Toxin-1 (TSST-1)** \\ \end{tabular} & \
\begin{tabular}{c} **Toxin 3** \\ _Staphylococcus aureus_ \\ **Enterotoxin Type A** \\ \end{tabular} &
\begin{tabular}{c} **Toxin 4** \\ _E. coli_ **Shiga Toxin 2 (One** \\ **Subunit A and Five** \\ **Subunits B)** \\ \end{tabular} \\ \hline \(\lambda_{D}\) (m) & \(1.79\times 10^{-5}\) & \(1.71\times 10^{-5}\) & \(3.51\times 10^{-5}\) & \(5.41\times 10^{-6}\) \\ \hline \(C_{M}\left(\frac{\text{mol}}{\text{m}^{3}}\right)\) & & \(56\times 10^{-9}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Calculation of \(\lambda_{D}\) per toxin. \(\lambda_{D}\): the diffuse layer width in the linear approximation which applies only when the applied potential is very small (it is a characteristic constant). \(C_{M}\): the toxin concentration at the middle of the duct after the electric potential application. If \(c_{D}\) is expressed as \(c_{D}=\frac{\xi}{\lambda}\), where \(\lambda\) represents the thickness of the effective capacitor of the diffuse layer, we can find the ratio:  
\[\frac{\lambda}{\lambda_{D}}=\left[\cosh\hbr{\frac{zF}{2 RT}\varphi_{s}}\right]^{-1} \tag{14}\]  
Figure 6 presents the change of \(\frac{\lambda}{\lambda_{D}}\) as a function of \(\varphi(0)\) (Equation (13)) for \(C_{M}=56\times 10^{-9}\frac{\text{mod}}{\text{m}^{3}}\) for the different toxins. As it is already mentioned, \(\lambda_{D}\) is the diffuse layer width in the linear regime, calculated for each toxin in Table 2. The physical significance of \(\lambda\) is that it indicates the effective width of the area where the toxins are cornered with the application of the electric field.