**3. Mathematical Framework**

#### *3.1. Capacitance Estimation*

In parallel with the development of the super-ENCs manufacturing procedures, analytical calculations have also been carried out to estimate the electrical properties that can be expected from these devices. Special attention has been paid to magnitudes as the capacitance, the behavior of the phase, and the impedance module when an AC current is applied, as well as the current flowing through the device when in use.

To estimate the capacitance of the device, it is essential to know the symmetry of the substrate on which the layers that make up the super-ENC are deposited. It is considered that inside each one of the NAAM's pores, a capacitor of cylindrical symmetry is generated, all of them with similar characteristics because of the geometrical uniformity of the NAAM. This single capacitor is connected in parallel with the six single capacitors of cylindrical symmetry present in the six adjacent pores, as a result of the spatial pores distribution with hexagonal symmetry. The total capacitance of the device is then the sum of the capacitance of all the individual capacitors present in the NAAM, as they are connected in parallel with each other. In Figure 6, a diagram of the cross-section of the capacitor generated within a pore can be seen. Three parts can be clearly differentiated according to the geometry, each of them associated with a type of capacitor. In the upper part, a flat symmetry capacitor appears; the intermediate part corresponds to a capacitor of cylindrical symmetry; and the lower part, the

bottom of the pore, is associated with a capacitor of hemispherical symmetry. From the calculations reported in the work of [5], where the capacitance provided by each of the parts is calculated, it follows that the main contribution to the total capacitance of the super-ENC comes from the cylindrical part of the pores. Taking into account that the pores of the NAAMs used in this work have a height of 1.2 μm, it is estimated that 95% of the capacitance of the super-ENC comes from the cylindrical part, so the contributions of the top flattened and bottom hemispherical parts are considered negligible.

**Figure 6.** Schematic cross-section of the supercapacitor conductor/dielectric/conductor (C/D/C) structure across a NAAM's pore. All the sections that make up the device are represented, starting from the aluminum substrate on which the pores are grown. Between the electrodes (TE and BE), the layers forming the dielectric material can be seen, representing those corresponding to the triple layer capacitor in this figure. The yellow color represents the SiO2 layers and the orange represents the one of TiO2, indicating their relative permittivity (εR1 and εR2) and thickness (a, b, c, and d). Note that for the single layered capacitor, instead of three dielectric layers, there would be only one, occupying the same space as the sum of the three layers.

One of the innovations presented in this work relays on the estimation of the capacitance for triple dielectric layer capacitors (SiO2/TiO2/SiO2) instead of single layer capacitors. This fact constitutes the main contribution of this work with respect to the calculations exposed in previous literature [5], where only single layer capacitors are treated. The capacitance expression for the cylindrical part is then slightly different, as shown in Equation (2), where *a*, *b*, *c*, and *d* represent the radii of the different dielectric layers with respect to the axis of symmetry of the cylinder. Therefore, the interdistance *d* − *c* corresponds to the thickness of the internal layer, *c* − *b* to that of the intermediate layer, and *b* − *a* to that of the outer layer of dielectric. Note that *a* is the distance from the axis of the cylinder to the outer layer of dielectric, a quantity that does not depend on the thickness of the TE. On the other hand, *r*<sup>p</sup> − *d* is the thickness related to the BE, but this one does not influence the capacitance calculations. As the internal and external dielectric layers are formed by the same material, two values of relative permittivity come into play, εr1 for the inner and outer layers (SiO2) and εr2 for the intermediate layer (TiO2). In the case of the super-ENC formed by a single layer (Al2O3), the capacitance calculation for a single pore is simpler (Equation (3)), obtaining an expression similar to that shown in the work of [5]. For the single layer capacitor, the thickness and relative permittivity of the dielectric layer is given by *d* − *a* and εr, respectively.

$$C = \frac{2\pi\varepsilon\_0\hbar}{\frac{1}{\varepsilon\_{r\_1}}\ln\left(\frac{db}{dt}\right) + \frac{1}{\varepsilon\_{r2}}\ln\left(\frac{c}{b}\right)}\tag{2}$$

$$\mathcal{C} = \frac{2\pi\varepsilon\_0\varepsilon\_r h}{\ln(d/a)}\tag{3}$$

To account for the total capacitance of the super-ENC, the capacitance density is usually calculated, that is, the normalized capacitance per unit area of the NAAM. It is necessary, therefore, to know the number of pores per unit area (σ) that the NAAMs present, which is given by Equation (4), where the hexagonal symmetry of the membranes is also taken into account.

$$
\sigma = \frac{2}{\sqrt{3}R\_{\text{CC}}^2} \tag{4}
$$

Then, the capacitance density of supercapacitors can be obtained as the result of multiplying Equations (4) and (2) or (3) (for tri-layered or single-layered dielectric material, respectively). By introducing *a, b, c*, and *d* distances according to the thicknesses of the respective dielectric layers (see Table 2), the geometrical parameters of the NAAMs, *R*CC, and *h* (see Figure 1 in the Manufacturing section), as well as the relative permittivity values, εr, (which are shown in Table 1), it is possible to estimate the capacitance densities for the manufactured super-ENCs (Table 5).

Because manufactured NAAMs have a surface area of 0.7 cm2, the expected capacitances for super-ENCs would be around of 6.7 and 14.8 μF for the triple and single dielectric layer capacitors, respectively.

**Table 5.** Distances from the pore axis to the different dielectric layers and estimation of the capacitance density for the manufactured devices.

