*3.1. Gouy–Chapmann–Stern Model*

The GCS model is the simplest extension of the PB theory in accounting for the effect of the finite size of ions. As illustrated Figure 5, it conceptualizes the structure of EDLs near the electrode surface as being composed of two layers: (i) a compact Stern layer where the ions are immobile and strongly adsorbed to the electrode surface, with a thickness corresponding to the closest approach of hydrated ions to the electrode surface; (ii) a diffuse layer where thermal motion causes the ions to be spread out in space.

It follows that no free charges exist in the Stern layer, while the distribution of ions in the diffuse layer can still be described by the Boltzmann equation, i.e.,

$$n\_i = n\_{i, \infty} \exp\left(-\frac{z\_i e \psi}{k\_B T}\right) \tag{3}$$

where *ψ* is the electric potential (V), *ni*,<sup>∞</sup> is number density (1/m3) of the *i*th species at the bulk solution, e is the elementary charge (C), *zi* is the valence of the *i*th species and *kB* is the Boltzmann constant (J/K), respectively.

The distribution of electric potential can, therefore, be described by the Poisson equation as

$$\nabla \times (\varepsilon\_0 \varepsilon\_r \nabla \psi) = -e \sum\_i z\_i n\_i \tag{4}$$

where *ε*<sup>0</sup> and *ε<sup>r</sup>* are the free space permittivity and the relative permittivity of the electrolyte solutions, ∇ is the divergence operator.

As a result, the GCS model can be expressed as:

$$\nabla^2 \psi = \begin{cases} 0 & \text{in Stern layer} \\ -\frac{c}{\varepsilon\_0 \varepsilon\_r} \sum z\_l n\_{i,\infty} \exp\left(-\frac{z\_l r \psi}{k\_B T}\right) & \text{in Diffuse layer} \end{cases} \tag{5}$$

subject to the following boundary conditions:

$$\left.\psi\right|\_{x=0} = \psi\_s \tag{6}$$

$$\left.\psi\right|\_{x=\delta^-} = \left.\psi\right|\_{x=\delta^+} \tag{7}$$

$$\left. \frac{d\psi}{d\mathfrak{x}} \right|\_{\mathfrak{x}=\delta^{-}} = \left. \frac{d\psi}{d\mathfrak{x}} \right|\_{\mathfrak{x}=\delta^{+}} \tag{8}$$

$$\left.\psi\right|\_{x\to\infty} = 0\tag{9}$$

where *x* is the normal distance from the electrode surface, *δ* is the thickness of the Stern layer and *<sup>ψ</sup><sup>s</sup>* is the surface potential, ∇<sup>2</sup> is the Laplace operator.

The GCS model, as given above, is quite general in that it can be applied to electrode particles with any geometry immersed in any electrolyte solution and can readily be solved numerically to obtain the relation between the surface potential *ψ<sup>s</sup>* and the surface charge *σs*. When used in practice, however, it is commonly assumed that the particle is of planar geometry and that the electrolyte is symmetric. For this special case, the analytical solution to the GCS model can be obtained to give the profile of electric potential *ψ* as a function of *x*,

$$\psi = \begin{cases} \begin{array}{c} \psi\_{\sf s} - \Delta \psi\_{\sf St} \times \text{x} / \delta \\ \frac{4k\_B T}{\varepsilon} \tanh^{-1} \left( \frac{\varepsilon \Delta \psi\_{\sf d}}{k\_B T} \right) \exp\left( \frac{\mathbf{x} - \delta}{\lambda\_D} \right) & \text{in Diffuse layer} \end{array} \tag{10}$$

with the Debye length *λ<sup>D</sup>* given by:

$$
\lambda\_D = \sqrt{\frac{\varepsilon\_0 \varepsilon\_r k\_B T}{2c^2 n\_{i,\infty}}} \tag{11}
$$

where Δ*ψSt* is the electric potential drop across the Stern layer, Δ*ψSt* = *ψ<sup>s</sup>* − *ψd*, Δ*ψ<sup>d</sup>* is the electric potential difference across the diffuse layer, Δ*ψ<sup>d</sup>* = *ψ<sup>d</sup>* − *ψ*<sup>∞</sup> and *ψ<sup>d</sup>* and *ψ*<sup>∞</sup> are the electric potential at the outer Stern plane and in the bulk solution, respectively.

It follows from Gauss' law that the surface charge *σ<sup>s</sup>* can be related to both Δ*ψSt* and Δ*ψ<sup>d</sup>* as

$$
\sigma\_{\rm s} = -\varepsilon\_0 \varepsilon\_r \left(\frac{d\psi}{d\mathbf{x}}\right)\Big|\_{\mathbf{x}=0} = -\varepsilon\_0 \varepsilon\_r \frac{\Delta\psi\_{\rm St}}{\delta} = \sqrt{8n\_{\rm cs}\varepsilon\_0 \varepsilon\_r k\_B T \sin\mathbf{h}\left(\frac{\mathbf{z}\varepsilon\Delta\psi\_d}{2k\_B T}\right)}\tag{12}
$$

The total differential capacitance of the double layer, *C*, can then be given by

$$\frac{1}{C} = \frac{1}{C\_{St}} + \frac{1}{C\_d} \tag{13}$$

where *Cst* and *Cd* are the differential capacitances to the Stern layer and the diffuse layer, respectively, with

$$\mathbf{C}\_{St} = -\frac{d\sigma\_s}{d\Delta\psi\_{St}} = \frac{\varepsilon\_0 \varepsilon\_r}{\delta} \tag{14}$$

and

$$\mathcal{C}\_{d} = -\frac{d\sigma\_{s}}{d\Delta\psi\_{d}} = \frac{\varepsilon\_{0}\varepsilon\_{r}}{\lambda\_{D}}\cos\mathrm{h}\left(\frac{ze\Delta\psi\_{d}}{2k\_{B}T}\right) \tag{15}$$

As a result, the surface charge density *σ<sup>s</sup>* from Equation (12) can also be written as a product of Δ*ψSt* and *CSt*, i.e.,

$$
\sigma\_s = -\Delta \psi\_{St} \times \mathbb{C}\_{St} \tag{16}
$$

#### *3.2. Modified Poisson–Boltzmann–Stern Model*

The GCS model accounts for the effect of the finite size of ions only in the Stern layer but treats the ions in the diffuse layer still as point-charges. In the case of high ion concentration and high surface potential, however, the interfacial region can be largely enriched in counterions to the extent that the point charge hypothesis for the EDL structure leads to unrealistically high counterion concentrations in the vicinity of the solid/solution interface [34]. This fact means a non-negligible role of the size of the ions, even in the diffuse layer, and therefore, as illustrated in Figure 5, a maximum ion concentration must exist corresponding to the closed packing of ions. As a result, the model for the diffuse layer should properly be modified to address the non-ideal behavior of ions therein. This leads Bikerman [34–36] to arrive at, by means of the approximate "free volume" approach:

$$m\_i = \frac{n\_{i,\infty} \exp\left(-\frac{z\_i c \psi}{k\_B T}\right)}{1 + \nu \sum\_k n\_{k,\infty} \left[\exp\left(-\frac{z\_k c \psi}{k\_B T}\right) - 1\right]}\tag{17}$$

where *ν* has the meaning of average excluded volume per ion.

When combined with the Poisson equation and consideration of a Stern layer, the MPBS model can be written for a (*z*:*z*) symmetric electrolyte as,

$$\nabla^2 \psi = \begin{cases} 0 & \text{in Stern layer} \\ -\frac{\varepsilon}{\varepsilon\_0 \varepsilon\_r} \frac{2zu\_\infty \sin \hbar \left(\frac{x\phi}{k\_B T}\right)}{1 + 2\nu \sin \hbar^2 \left(\frac{x\phi}{2k\_B T}\right)} & \text{in Diffuse layer} \end{cases} \tag{18}$$

with

$$\mathbf{v} = 2d^3 n\_{\infty} \tag{19}$$

where *d* is the spacing of counterions near a highly charged surface, and it is unnecessarily the diameter of the counterions. One may think of it as a cutoff [36] for the unphysical divergences of PB theory and could include at least a solvation shell (ion–ion correlations could effectively increase it further).

Equation (18) should also be subject to the same boundary conditions of Equations (6)–(9) as the GCS model, and therefore it is also convenient to be solved numerically. However, for a planer electrode surface, an analytical solution to the surface charge density *σ<sup>s</sup>* is available for a symmetric electrolyte and it can be written as,

$$\sigma\_{\rm s} = -2z e n\_{\rm cs} \lambda\_D \sqrt{\frac{2}{v} \ln \left[ 1 + 2v \sin \text{h}^2 \left( \frac{z e \Delta \psi\_d}{2k\_B T} \right) \right]} \tag{20}$$

The relation between *σ<sup>s</sup>* and Δ*ψSt*, as given in Equation (16), also holds in the MPBS model.
