*4.3. Modified Donnan Model*

In contrast to the GCS and MPBS models, the mD model considers that the micropores of activated carbon particles have a size comparable to the EDL thickness and even to that of hydrated ions, leading to EDLs overlap and a constant diffuse layer potential. The total charge in electrode at equilibrium can, therefore, be calculated as:

$$Q = -\frac{1}{2} \times \rho \times v\_{mi} \tag{31}$$

where *vmi* is the micropore volume per unit electrode mass (cm3/g).

Using this expression for *Q* and Equation (30) for *V*, the *Q*–*V* curves of the CDLE cell under different electrolyte concentrations can conveniently be obtained in the mD model. The unknown parameters involved are, however, version-dependent. In the standard mD model [29,30,38,39,45], where *μatt* is taken as a constant, the optimal values of *μatt*, *vmicro*, *CSt*,*vol*,0 and *α* are all in need of determination from the procedure of fitting the simulated *Q*–*V* curves to the measured data over all the NaCl concentrations of interest. The result shows when a nonlinear least square algorithm is used, *μatt* = 1.18, *vmi* = 0.35 cm3/g, *<sup>α</sup>* = 10.5 F·m3/mol2 and *CSt*,*vol*,0 = 2.1 × 108 F/m3. These values are in line with those suggested by other works [30,38,45], with a good agreement between the simulated and measured *Q*–*V* curves as a result, as shown in Figure 12.

The fact of *vmi* = 0.35 cm3/g suggests that about 90% of the micropore space is available for counterion adsorption. This is deemed to be much more reasonable than the result (∼70%) of the GCS and MPBS models imply. On the other hand, *CSt*,*vol*,0 = 2.1 × 108 F/m<sup>3</sup> is equivalent to *ε<sup>r</sup>* = ∼7.29 given a Stern layer thickness on the order of magnitude of hydrated ions of Na+. This is also reasonable, as Booth correction of dielectric permittivity (see Figure 7) suggests. Therefore, the mD model is physically much more plausible than both GCS and MPBS models.

**Figure 12.** Equilibrium electrode charge *Q* versus applied voltage *V* for different values of NaCl solution. Lines refer to the results of mD model (*<sup>a</sup>* = 10.5 F·m3/mol2, *CSt*,*vol*,0 = 2.1 <sup>×</sup> 108 F/m3, ߤୟ୲୲ = 1.18, *vmi* = 0.35 cm3/g), marks refer to the experiment data and error bars of the experimental data are indicated by horizontal lines through the marked data points.

In the improved mD (i-mD) model [39], the excess chemical potential, *μatt* is related to the total concentration of all ions in the micropores as given in Equation (25). The energy parameter E should now be determined a priori, instead of *μatt*. Using the same fitting procedure as discussed above, we found that the optimal values of the fitting parameters are: *<sup>E</sup>* = 436.7 kT mol/m3, *CSt*,*vol*,0 = 2.06 × 108 F/m3, *<sup>α</sup>* = 13.6 F·m3/mol2 and *vmi* = 0.364 cm3/g. The value of *vmi* = 0.364 cm3/g suggests that the availability of micropore volume for storing the counterions is ∼94%. *CSt*,*vol*,0 = 2.06 × <sup>10</sup><sup>8</sup> F/m3 corresponding to *ε<sup>r</sup>* = ~7.17 also implies a reasonable Stern layer thickness. The difference between mD and i-mD models is, therefore, mainly on the *CSt*,*vol*,0 values. As shown in Figure 13, the decrease of *CSt*,*vol* with increasing volume charge in both mD models follow essentially the same pattern. As a result, *ε<sup>r</sup>* also decreases but only slightly. It becomes ~6.77 and ~6.49 at *<sup>ρ</sup>* = 1.2×108 C/m3 in the mD and i-mD model, respectively.

With the optimal parameters, it is seen from Figure 14 that the simulated *Q*–*V* curves by i-mD model agrees almost entirely with the experimental data, much better than the results of the other models (see Figures 7–12).

To facilitate the comparison of different EDL models, we summarize in Table 2. the physical parameters involved in the models and the optimal values obtained from the fitting procedures. As discussed above, the parameter values of both mD and i-mD models are not only reasonable but also roughly the same. However, as clearly seen from the results shown in Figures 12 and 14, the i-mD model is superior to the mD model in reproducing the dependence of *Q* on *V* under different electrolyte concentrations.

**Figure 13.** Stern layer capacity *CSt*,*vol* as a function of surface charge density *ρ*.

**Figure 14.** Equilibrium electrode charge *Q* versus applied voltage *V* for different values of NaCl solution. Lines refer to the results of i-mD model (*<sup>a</sup>* = 13.6 F·m3/mol2, *CSt*,*vol*,0 = 2.06 <sup>×</sup> <sup>10</sup><sup>8</sup> F/m3, *E* = 436.7 kT mol/m3, *vmi* = 0.36 cm3/g), marks refer to the experiment data and error bars of the experimental data are indicated by horizontal lines through the marked data points.


**Table 2.** Optimal parameters values for different theoretical models in the present paper.

\*: with Booth correction.

By contrast, the optimal parameters of the GCS and the MPBS models are physically unreasonable, especially about the Stern layer thickness. As seen in Table 2, even the minimum d value of 4.19 nm is still far greater than the hydrated radius [44] of Na+ ions, in contradiction with the physical explanation of the Stern layer. It is mainly this finding that makes us believe the applicability of the GCS and MPBS models is questionable.
