*3.3. Modified Donnan Model*

When dealing with the experimental results of CDLE processes, the application [28] of both GCS and MPBS models implicitly assume that the characteristic pore thickness is much larger than the Debye screening length so that the pore space is mostly filled with quasi-neutral electrolyte, exchanging ions with a charged, thin double-layer "skin" on the electrode matrix. This is, however, far from realistic because the activated carbon particles are themselves porous, presenting a very large specific surface inside the small micropores (≤2 nm). Therefore, as schematically shown in Figure 6, the free space between different carbon particles filled with electroneutral solution constitutes a macro-porosity that serves as a path for salt and charge transport, whereas the micropores store ionic charge in their EDLs. This fact implies that both GCS and MPBS descriptions of the EDL are not valid inside the micropores of activated carbon particles, which have a size comparable to the EDL thickness and even to that of hydrated ions, leading to EDLs overlap and other complications, such as the observed exceptionally large values of the capacitance.

**Figure 6.** Schematic view of the structure of porous electrode.

To tackle this problem, Biesheuvel et al. [37] combined a modified Donnan description of the diffuse layer together with a charge-free Stern layer to determine the voltage drop at the carbon/solution interface inside the micropores. The main assumption of the mD approach is that the diffuse layer potential inside the micropores is constant, as illustrated in Figure 5, and it is controlled by the concentration of ions in the macropores of the electrode matrix, i.e., one can write [37]

$$n\_{i,mi} = n\_{i,\infty} \exp\left(-\frac{z\_i e \Delta \psi\_d}{k\_B T} + \mu\_{att}\right) \tag{21}$$

where the subscript "mi" emphasizes that it applies only to micropores, Δ*ψ<sup>d</sup>* is now known as the Donnan potential and *μatt* is an excess chemical potential that quantifies the chemical attraction between ions and carbon material of the electrodes [37].

This allows one to introduce the concept of volumetric charge density as the number of charges removed from solution per unit micropore volume, and in the case of a symmetric binary *z* : *z* electrolyte, it is given by [31]

$$\rho = -2zen\_{\infty} \exp(\mu\_{att}) \times \sin \hbar \left(\frac{ze\Delta\psi\_{\rm d}}{k\_B T}\right) \tag{22}$$

and

$$
\rho = -\mathbb{C}\_{\text{St,vol}} \Delta \psi\_{\text{St}} \tag{23}
$$

where *CSt*,*vol* is the volumetric capacitance (F/m3) of the Stern layer, and it was suggested to be quantified empirically by [38,39]

$$\mathcal{C}\_{\text{St,vol}} = \mathcal{C}\_{\text{St,vol,0}} + \alpha \times \left(\frac{\rho}{F}\right)^2 \tag{24}$$

with both *CSt*,*vol*,0 and *<sup>α</sup>* (F·m3/mol2) being determined by fitting the mD model to the experimental data.

One key of the mD model is to describe the excess chemical potential properly, *μatt*. For simplicity, it was generally taken as a constant irrespective of the specific type of ions [30,38]. Although this assumption makes the mD model works well for some cases, it cannot describe the experimental data in a range of bulk salt concentrations simultaneously [37]. An improved modified Donnan model (i-mD model) was developed [39] to rectify this problem by relating *μatt* with the micropore total ion concentration *cions*,*mi*, based on the theory of image forces, to give

$$
\mu\_{att} = \frac{E}{c\_{ions\_{\rho}mi}} \tag{25}
$$

with the energy parameter (kT mol/m3), *E*, defined as

$$E = z^2 \times k\_B T \times \lambda\_B \times d\_p^{-4} \tag{26}$$

where *λ<sup>B</sup>* is the Bjerrum length, *λ<sup>B</sup>* = *e*2/4*πε*0*εrkBT*, at which the bare Coulomb energy of a pair of ions is balanced by thermal energy (*λ<sup>B</sup>* = 0.72 nm in water at room temperature), and *dp* is the size of micropore.
