*4.2. Modified Poisson–Boltzmann–Stern Model*

The MPBS model bears the same conceptualization on the EDL geometry as the GCS model but accounts for the effect of finite-size of ions. We can, therefore, still use Equations (29) and (30) to calculate *Q* and *V* in the simulations. As shown in Figure 10, the agreement between the calculated and measured *Q*–*V* curves is nearly the same as the GCS model does, with *CSt* = 0.141 F/m2 and *Seff* = 580.88 m2/g when *ε<sup>r</sup>* is fixed at 78.5. The minor difference between the *CSt* value of the two models also suggests that the MPBS model breaks down in the regime of experimental interest.

**Figure 9.** Equilibrium electrode charge Q versus applied voltage V for different values of NaCl solution. Lines refer to the results of the GCS model with Booth correction (*δ* = 4.2 nm, *Seff* = 540.25 m2/g), marks refer to the experiment data and error bars of the experimental data are indicated by horizontal lines through the marked data points.

**Figure 10.** Equilibrium electrode charge *Q* versus applied voltage *V* for different values of NaCl solution. Lines refer to the results of MPBS model (*CSt* = 0.141 F/m2, *Seff* = 580.88 m2/g, *ε<sup>r</sup>* = 78.5, *d* = 0.75 nm), marks refer to the experiment data and error bars of the experimental data are indicated by horizontal lines through the marked data points.

Not surprisingly, when combined with Booth correction, the MPBS model does not move the needle at all. It gives *δ* = 7.3 nm, *Seff* = 857.79 m2/g and the spacing of counterions *d* = 1.42 nm, with an acceptable agreement between the calculated and measured *Q*–*V* curves, as shown in Figure 11. As a result, accounting for the variation of *ε<sup>r</sup>* with the electric field in the MPBS model does not help to rectify the problems associated with the model in the cases where the assumption of thin double-layer "skin" on the electrode matrix is not valid.

**Figure 11.** Equilibrium electrode charge *Q* versus applied voltage *V* for different values of NaCl solution. Lines refer to the results of MPBS model with Booth correction (*δ* = 7.3 nm, *Seff* = 857.79 m2/g, *d* = 1.42 nm), marks refer to the experiment data and error bars of the experimental data are indicated by horizontal lines through the marked data points.
