*3.4. Booth Correction of Dielectric Permittivity*

In most applications of the GCS and MPBS models, the solvent dielectric permittivity was assumed to be a constant or was treated as a fitting parameter. However, it is known that with the increase in the electric field, the relative permittivity of solvent *ε<sup>r</sup>* shows a decreasing trend. The reason for this is that under a large electric field condition, electrolyte molecules become highly oriented, which results in poor capability of providing polarization [40]. To account for this effect of dielectric saturation into EDL models, Booth [41,42] derived the following equations to calculate the relative electrolyte permittivity under the local electric field condition as:

$$\varepsilon\_{\rm r}(E) = \begin{cases} m^2 + \left(\varepsilon\_{\rm r}(0) - m^2\right) \frac{3}{\rm PE} \left[\coth\left(\beta E\right) - \frac{1}{\rm PE}\right] & \text{for } E \ge 10^7 \text{ V/m} \\\quad \varepsilon\_{\rm r}(0) & \text{for } E < 10^7 \text{ V/m} \end{cases} \tag{27}$$

with

$$
\beta = \frac{5\mu}{2k\_B T} \left( m^2 + 2 \right) \tag{28}
$$

where *E* = |−∇*ψ*| is the norm of the local electrical field vector, *εr*(0) is the relative permittivity at zero electric field, *m* is the index of refraction of the electrolyte at zero electric field frequency, *μ* is the dipole moment of the solvent molecule, and in the case of water, *μ* = 1.85 D (Debye).

For aqueous binary symmetric electrolytes at room temperature (*T* = 298 K), one may set [40,43]: *<sup>ε</sup>r*(0) = 78.5, *<sup>m</sup>* = 1.33 and *<sup>β</sup>* = 1.41 × <sup>10</sup>−<sup>8</sup> m/V. The result for the relative permittivity as a function of the electric field is shown in Figure 7. It suggests that *ε<sup>r</sup>* may change significantly from 78.5 in the bulk solution to 1.79 near the electrode surface in the cases where the electric field is very strong.

**Figure 7.** The relative permittivity *εr*(*E*) as a function of electric field.

The importance of the Booth correction of dielectric permittivity is, therefore, that it may be used to account for the non-ideal behavior (or excess chemical potential) of ions in EDLs in a different way than MPBS and mD models and that it can be combined with either GCS or MPBS model for the performance assessment of CapMix processes.

#### **4. Experimental Results and Model Applications**

The key property of the CDLE cells is the dependence of the equilibrium electrode charge *Q* on the applied voltage *V* under different electrolyte concentrations, also known as the *Q*–*V* curves. The applicability of different EDL models can, therefore, be justified by comparing the simulated *Q*–*V* curves with the results of single-pass experiments.
