*4.1. Gouy–Chapmann–Stern Model*

As discussed above, the GCS model is usually applied to the cases where the Debye screening length of the EDL, *λD*, is much less than the characteristic pore thickness *Hp* so that a planer electrode surface could effectively be defined. As a result, *Q* (C/g) can simply be evaluated by multiplying *σ<sup>s</sup>* with the specific electrode area *Seff* (m2/g) as:

$$Q = \frac{1}{2}\sigma\_{\mathfrak{s}} \times S\_{eff} \tag{29}$$

The corresponding voltage drop across the cell can, by assuming it to be evenly distributed on the two electrodes, relate to Δ*ψSt* and Δ*ψ<sup>d</sup>* directly as:

$$V = 2(\Delta\psi\_{St} + \Delta\psi\_d) \tag{30}$$

With these definitions, the GCS model can readily be applied to the CDLE cell to obtain the *Q*–*V* curves under different electrolyte concentrations. Since it is commonly assumed in the GCS model that *ε<sup>r</sup>* = *εr*(0), the unknown parameters of the specific system only involve *Cst* and *Seff* . As a result, we may fix *ε<sup>r</sup>* = 78.5 at *T* = 298 K and evaluate *Cst* and *Seff* simultaneously by fitting the simulated *Q*–*V* curves to the measured data over all the NaCl concentrations of interest by using a nonlinear least square algorithm, supplemented with suitable lower and upper bounds. The result gives *CSt* = 0.131 F/m2 and *Seff* = 619.46 m2/g. At first glance, a good agreement between the calculated and measured *Q*–*V* curves, as shown in Figure 8 and the fact that *Seff* = 619.46 m2/g is comparable with the findings of reported results [19,22,23] seems to substantiate the rationale of *Cst* and *Seff* values. However, further analysis suggests that *CSt* = 0.131 F/m2 corresponds to a Stern layer thickness of 5.3 nm. This is far greater than the hydrated radius [44] of Na+ ions and is therefore unreasonable, meaning that an arbitrary setting of *CSt* = 0.1 F/m2, as shown by previous work [21,22], is also problematic.

**Figure 8.** Equilibrium electrode charge Q versus applied voltage V for different values of NaCl solution. Lines refer to the results of GCS model (*CSt* = 0.131 F/m2, *Seff* = 619.46 m2/g, *ε<sup>r</sup>* = 78.5), marks refer to the experiment data and error bars of the experimental data are indicated by horizontal lines through the marked data points.

On the other hand, it is noted from Table 1. that the characteristic pore thickness *Hp* is on the order of magnitude of 0.43 nm, whereas the Debye screening length of the EDL is about 0.4 nm at a bulk concentration of 600 mM. This suggests that *Hp* is always smaller than *λ<sup>D</sup>* in all the cases studied, and therefore violates the assumption of thin double-layer "skin" on the electrode matrix. As a result, the GCS model is deemed to be not applicable, especially when the NaCl concentration is small.

If combined with Booth correction of dielectric permittivity, the GCS model gives even worse agreement, as shown in Figure 9, with *δ* = 4.2 nm and *Seff* = 540.25 m2/g. This suggests that accounting for the variation of *ε<sup>r</sup>* with the electric field does not remedy the inherent problem of the GCS model, still making the parameters physically meaningless, and therefore should also be abandoned in the interpretation of the experimental results.
