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# Complex conductivity as a tool to investigate the electrical behavior between graphene oxide and reduced graphene in supercapacitors properties

Dalal Hilmi, Soumia Zaim, Abdehadi Mortadi, Ibtissam Sabir, Mohamed Monkade, Rachid Nmila, Alina Violeta Ursu, Christophe Vial, Philippe Michaud, Halima Rchid, et al.

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Dalal Hilmi, Soumia Zaim, Abdehadi Mortadi, Ibtissam Sabir, Mohamed Monkade, et al. Complex conductivity as a tool to investigate the electrical behavior between graphene oxide and reduced graphene in supercapacitors properties. Results in engineering, 2024, 23, pp.102673. 10.1016/j.rineng.2024.102673. hal-04858768

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# Complex conductivity as a tool to investigate the electrical behavior between graphene oxide and reduced graphene in supercapacitors: Correlation between the electrical properties

Dalal Hilmi<sup>a,b,*</sup>, Soumia Zaim<sup>a</sup>, Abdehadi Mortadi<sup>a</sup>, Ibtissam Sabir<sup>b</sup>, Mohamed Monkade<sup>a</sup>, Rachid Nmila<sup>b</sup>, Alina Violeta Ursu<sup>c</sup>, Christophe Vial<sup>c</sup>, Philippe Michaud<sup>c</sup>, Halima Rchid<sup>b</sup>, Reddad El Moznine<sup>a,**</sup>

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## ARTICLE INFO

**Keywords:**

Graphene oxide

Reduced graphene oxide

Conductivity

Relaxation time

Equivalent circuit

## ABSTRACT

The storage mechanisms of solid-solid supercapacitors based on graphene oxide (rGO/GO/rGO) were explored by Zhang Q et al. (Zhang et al., 2014) [1], using Density Functional Theory (DFT) and Impedance Spectroscopy (EIS) to elucidate the unusual capacitive behavior of GO sandwich films. Their study revealed alternating charged layers without diffusion zones. Analysis of complex impedance (Z*) data, ranging from 10<sup>-3</sup> to 3.10<sup>6</sup> Hz, was conducted using an equivalent circuit, though this was limited to Nyquist plot data.

To further investigate charge storage mechanisms, the current study delved deeper into complex impedance (Z*) and conductivity (σ*). A theoretical approach identified relaxation processes in the imaginary parts of (Z*) and (σ*) across frequencies. Extrapolation up to 3.10<sup>9</sup> Hz and deconvolution confirmed three distinct relaxation processes in Z* and σ*, similar to Cole-Cole relaxation, which describes Maxwell-Wagner-Sillars (MWS) relaxation. This reflects local electrical interactions between active sites on molecular chains and counterions, creating induced dipolar moments or diffusion processes.

The identified relaxations at low, medium, and high frequencies correspond well with DFT results, attributed to specific regions: graphene oxide (GO), the (rGO/GO) interface, and reduced graphene (rGO). Key parameters extracted from each relaxation, such as relaxation time, ionic strength, and conductivity values at different frequencies, showed good correlation. This approach, based on deep conductivity (σ*) analysis, effectively highlighted the charge conduction and storage mechanisms in rGO/GO/rGO supercapacitor films.

## 1. Introduction

The scientific community has experienced a notable surge in fasci-
nation with graphene and its derivatives following its groundbreaking
isolation in 2004 by Geim A [1]. Graphene, characterized by a honey-
comb lattice of carbon atoms arranged in a hexagonal structure, exhibits
a two-dimensional (2D) configuration. This remarkable material,
available in both oxidized form known as Graphene Oxide (GO), and
reduced form known as reduced Graphene Oxide (rGO), exhibit excep-
tional electrochemical properties, making it a promising candidate for
supercapacitor applications. The versatility of graphene and its related

materials is evidenced by their successful utilization in a wide array of
fields, including energy storage, solar cells, electronics, and electro-
chemistry [2–10]. The exponential growth in interest surrounding gra-
phene signifies its immense potential to revolutionize various
technological domains.

Graphene oxide (GO) is an oxidized form of graphene, a two-
dimensional carbon sheet, that introduces oxygenated functional
groups (hydroxyl, carboxyl, and epoxide groups) onto its surface
through graphene oxidation [11]. Graphene oxide exhibits relatively
lower conductivity compared to graphene. Bassically, GO has been
experimentally proven to function as a solid electrolyte for

<footnote><sup>a</sup> Laboratory of Physics of Condensed Matter (LPMC).Chouaib Doukkali University, El Jadida, Morocco</footnote>

<footnote><sup>b</sup> Laboratory of Biotechnology and Valorization of Vegetal Resources.Chouaib Doukkali University, Morocco</footnote>

<footnote><sup>c</sup> Université Clermont Auvergne, CNRS, SIGMA Clermont, Institut Pascal, F-63000, Clermont-Ferrand, France</footnote>

<footnote>* Corresponding author. Laboratory of Physics of Condensed Matter (LPMC).Chouaib Doukkali University, El Jadida, Morocco.</footnote>

<footnote>** Corresponding author.</footnote>

<footnote>E-mail addresses: dalal.hilmi@gmail.com (D. Hilmi), elmoznine@yahoo.fr (R. El Moznine).</footnote>
supercapacitors by effectively trapping water and ions within the in-
terspaces of GO films, eliminating the need for foreign electrolytes [12].
On the other hand, reduced graphene oxide (rGO) is obtained by
reducing the oxygenated functional groups present on the graphene
oxide's surface, resulting in enhanced electrical conductivity, and
making it more suitable for electrochemical applications like super-
capacitors [13,14].

In recent years, electrochemical impedance spectroscopy (EIS) has become one of the most attractive diagnostic techniques due to its speed, precision, and low cost. To analyze complex electrochemical systems, such as supercapacitors based on graphene oxide (GO) films, it is essential to first understand the system as a whole. A comprehensive understanding allows for the construction of a physically coherent circuit model to interpret the impedance spectrum characteristics. EIS enables the characterization of complex impedance over a wide frequency range thus exploring relaxation processes in the electric double layer (EDL) on a sub-millisecond time scale [15]. For batteries and supercapacitors, frequencies beyond MHz enable EIS to identify different contributions of electrochemical phenomena to the internal resistance of devices and to determine components for improvement the increase of power density and to analysis the ionic transport properties in solid electrolytes [16,17]. Additionally, the distribution of relaxation times has been used to separate the overall impedance response into individual relaxation processes, offering a more detailed understanding of supercapacitor impedance spectra. This allows the differentiation between overlapping relaxation times, providing insights into the electrochemical behavior of graphene-based supercapacitors and improving the interpretation of complex impedance responses [18]. In this regard, impedance spectroscopy offers practical advantages for investigating graphene oxide sandwiched supercapacitors [19,20], including the utilization of deconvolution techniques to differentiate and quantify various relaxations, thereby facilitating the comprehension of mechanisms involving charge accumulation and release, ultimately enabling efficient energy storage and delivery [21].

The study carried out by Zhang et al. [12] focused on the application of impedance spectroscopy to characterize the charge transport properties of graphene oxide sandwich supercapacitors in the frequency range [10⁻³–3.10⁶ Hz]. The electrical circuit used to analysis the complex impedance Z*(ω) shows a good agreement with their experimental results.

The investigation revealed the captivating characteristics of semi-
conductor capacitors featuring a sandwich structure of rGO/GO/rGO.
These electrochemical devices exhibit atypical behaviors, displaying a
notably high specific capacity comparable to conventional electro-
chemical capacitors. However, their significant ion diffusion aligns them
with dielectric capacitors, suggesting the emergence of a third category
of capacitors. This promising convergence opens new avenues for
innovative applications in the field of energy storage.

The electrical circuit used for analyzing complex impedance Z*(ω) agreed well with experimental results. However, it should be noted that the analysis was carried out within a limited range of frequencies up to 3 MHz, which restricts the analysis of the complex conductivity σ*(ω) and the main characteristics related to the high relaxation processes beyond 3 MHz. In our recent work [22,23], the analysis of the complex impedance (Z*) as well as conductivity (σ*) of polysaccharide concentrations in aqueous solutions. This approach was very useful for investigating the electrostatic interactions between the active sites on molecular chains (e.g., hydroxyl, carboxyl groups) and counter ions. This study allowed us to determine the optimal concentration for the transition from the diluted to semi-diluted solution due to the entanglement between molecular chains.

Therefore, the current study aims to investigate charge conduction
and storage mechanisms in sandwiched graphene oxide-based super-
capacitors (rGO/GO/rGO) based on a further analysis of the complex
impedance (Z*) as well as the ionic conductivity (σ*). Our approach
focuses on extracting the key characteristics such as relaxation time,

ionic strength, and conductivity at different frequencies. The analysis of
these main electrical properties and their correlation are well estab-
lished and discussed for better understanding of the charge dynamics in
each specific region of (rGO/GO/rGO) capacitor films.

## 2. Simulation procedures

As mentioned earlier, impedance spectroscopy is a highly adaptable
and non-invasive approach that enables the exploration of diverse po-
larization characteristics within electrochemical systems. These include
the analysis of transport and charge transfer processes, interface phe-
nomena, diffusion, and conductivity.

For this reason, Zview. 2.2 software was employed to perform sim-
ulations based on the electrical parameters derived from the initial
equivalent circuit proposed by Qing Zhang et al. [12]. In the first step,
the frequency range [10⁻³–3.10⁶ Hz] was similar to that explored by
these authors in order to regenerate the complex impedance (Z*) data
provided. However, it was observed that this frequency range is very
limited, especially for the analysis of the complex conductivity (σ*).
Therefore, an extrapolation procedure up to 3.10⁹ Hz was performed in
order to further highlight the existence three relaxation processes.

3. Results and discussion

Before starting our analysis, it is worth to report few figures that can
be considered as the background from the previous study by Zhang. Q
et al. [12], which were investigated for the storage mechanisms of
supercapacitors structured as rGO/GO/rGO. Fig. 1(a and b) display
Nyquist plots spanning from 1 MHz to 3 MHz, showing two semicircles
indicating of the absence of diffusion processes, as evidenced by the lack
of a straight line at low frequencies typical in conventional electro-
chemical capacitors (ECs). Based on density functional theory (DFT)
results, the Charge Close-Packed (CCP) model was proposed to elucidate
the anomalous capacitive behavior of the GO sandwich film, empha-
sizing alternating positively and negatively charged layers devoid of
diffusion zones.

The electrical equivalent circuit shown in Fig. 1(b) successfully matched the complex impedance data in the Nyquist plot. It includes a series resistance (Rs) and three blocks of sub-circuits containing a parallel resistance with a constant phase element (R//CPE). Based on the results obtained in Fig. 2 for the charge distributions and molecular structures of water confined within the various layers, in addition to the deconvolution illustrated in the Nyquist plot, the authors [12] attributed: The series resistance (Rs) to the electrolyte resistance (Re). The contribution of each sub-circuit block to: the interlayer (rGO/rGO), the rGO/GO interface, and the GO/GO interlayer: The first; include the charge transfer resistance (R₁) in parallel with a constant phase element (CPE₁) employed to describe the non-ideal behavior of the capacitor of the interlayer rGO/rGO. The second block comprising a parallel combination of resistance (R₂) and constant phase element (CPE₂) was introduced to account for the rGO/GO interface. Lastly, the third block consisting of parallel R₃ and CPE₃ was included to represent the GO/GO interlayer. The circuit exhibited excellent agreement with the experimental data. The values of the fitting parameters are summarized in Table 1.

However, at this stage, it seems to state that the complex impedance Z* data recorded in the frequency range [10⁻³–3.10⁶ Hz] was investigated only in the Nyquist plot, where only the evolution of the imaginary part of the impedance (Z") versus the real part (Z') could be analyzed. In addition, the deconvolution procedure shown in the Nyquist plot was not illustrated very well.

Therefore, in the following section, we performed a simulation based
on the electrical parameters derived from the fitting with the equivalent
circuit in the same frequency range to regenerate the complex imped-
ance (Z*) data. Then, further investigations were performed on the
complex impedance (Z*) data in Nyquist plot as well as Bode plots.
<figure><img src="image_4.png" /><figcaption>**Fig. 1.** (a–b). (a) Nyquist plot from 1 mHz to 3 MHz and Molecular structures and charge distributions of the sandwiched GO-capacitor film (rGO/GO/rGO). (b) An equivalent circuit was used to fit the experimental data in the Nyquist plot.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Molecular structures and charge distributions. Molecular structures of water confined within (a) two rGO layers, (b) one GO and one rGO interface, and (c) two GO layers respectively. (d–e–f) Charges density for each specific region respectively along z-direction perpendicular to graphene plane under an external electrical field. (g) Sub-circuit corresponds to each specific region respectively.</figcaption></figure>

### 3.1. Analyses of the complex impedance Z*(ω)

The complex impedance data ($Z^*(\omega) = Z'(\omega) - iZ''(\omega)$) can be investigated by exploring various representations such as the evolution of the real $Z'(\omega)$ part and the imaginary parts $Z''(\omega)$ versus the angular frequency ($\omega$) known as Bode plots in addition to the evolution of $Z''(\omega)$ versus $Z'$ known as Nyquist plots. This kind of the analysis is crucial for understanding the electrical behavior, such as diffusion and relaxation processes [23]. Indeed, the diffusion process usually exhibits a straight line in low-frequency from Nyquist plots. In contrast, the relaxation process can be illustrated by the presence of a semi-circle in the Nyquist plot and simultaneously the presence of a maximum at a critical frequency in the evolutions of the imaginary part of the impedance $Z''(\omega)$

versus the angular frequency.

Fig. 3(a and b) shows the evolution of the complex impedance ($Z^*$) of the sandwiched graphene oxide-based supercapacitors (rGO/GO/rGO) in Nyquist (a) and Bode (b) plots.

From the Nyquist plot (Fig. 3 (a)), it can be seen that the trend and the evolution are similar to that obtained by Zhang, Q et al. [12] as it was shown in Fig. 1(a). The curve shows two depressed semicircles. The first is in the high-frequency region with a small depressed semicircle, whereas the second seems to be a very broad and asymmetric depressed semicircle in the medium and low-frequency regions. In addition, there is a significant discernible reduction in the diameter of these semicircles without the presence of a straight line in the low-frequency region [24].

The deconvolution procedure in the Nyquist plot offers a notable advantage by providing a clear visual representation of three semicircular arcs. Furthermore, the semicircle radius in the Nyquist plots serves as an indicator of conductivity and the charge-transfer resistance ($R_{CT}$) that governs the limiting process at the rGO/rGO, rGO/GO, or GO/GO interfaces [12]. Basically, a distinct trend becomes apparent when examining the $R_{CT}$ values in the Nyquist plots: $R_{CT}(GO/GO) > R_{CT}(rGO/GO) > R_{CT}(rGO/rGO)$. This demonstrates that the charge transfer resistance in the low-frequency range is larger than that in the high-frequency range. This observation suggests that oxygenated functionalities play a crucial role in influencing the dielectric response, leading to an increase in charge transfer resistance for (GO) layers relative to (rGO) layers.

From Fig. 3 (b), it can be observed that the evolution of the imaginary part ($Z''$) as a function of frequency simultaneously revealed the existence of two distinct regions in the frequency range indicated by the presence of two maxima: The first is located in a high-frequency region which is symmetric, whereas the second is located in the low-frequency region but it is very broad and asymmetric. This behavior could suggest the existence of another maximum in the medium-frequency region, which can be hidden by the clear maximum in the low-frequency region. More interestingly, the evolution of the real part ($Z'$) of the complex impedance ($Z^*$) versus frequency revealed also a change significant in the slope in the two different regions in the frequency region. This finding could reflect very well the importance to explore Bode plot representation indicating that the dynamic behavior could be revealed by the presence of three distinct peaks at specific frequencies. In addition, the evolutions of the complex impedance ($Z^*$) in Bode and Nyquist plots reflects the same behavior for the evolutions for the complex permittivity ($\epsilon^*$), which is usually used to describe the behavior of the

<table><caption>**Table 1**

Fitting parameters with the equivalent circuit.</caption><thead><tr><th rowspan="2">Electrolyte<br>Re (Ω)</th><th colspan="3">Interlayers rGO/rGO</th><th colspan="3">Interlayers rGO/GO</th><th colspan="3">Interlayers GO/GO</th></tr><tr><th>T<sub>1</sub> (F)</th><th>p<sub>1</sub></th><th>R<sub>1</sub> (Ω)</th><th>T<sub>2</sub> (F)</th><th>p<sub>2</sub></th><th>R<sub>2</sub> (Ω)</th><th>T<sub>3</sub> (F)</th><th>p<sub>3</sub></th><th>R<sub>3</sub> (Ω)</th></tr></thead><tbody><tr><td>11,19</td><td>7,3.10<sup>-8</sup></td><td>0,7574</td><td>377</td><td>5,3.10<sup>-4</sup></td><td>0,5025</td><td>1138</td><td>1.16.10<sup>-3</sup></td><td>0,8302</td><td>1459</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Fig. 3. (a–b). Complex impedance (Z*) of the sandwiched graphene oxide-based supercapacitors (rGO/GO/rGO). Nyquist (a) and Bode (b) plots.</figcaption></figure>

Cole-Cole relaxation process [25]:

$$ \varepsilon^*(\omega) = \varepsilon_{\infty} + \frac{\varepsilon_s - \varepsilon_{\infty}}{1 + (i\omega\tau_e)^p} \quad (1) $$

Where, εs and ε∞ correspond to the dielectric constants at low-frequency and high-frequency, respectively. The relaxation time is denoted as τe, and Δε represents the dielectric strength (Δε = εs·ε∞). The exponent parameter p ranges between 0 and 1.

For p close to 1, the Cole-Cole relaxation becomes equivalent to the Debye relaxation type. It is worth noting that this latter relaxation involves only a single relaxation time (τ).

For p lower than 1, this exponent is an indicator of the distribution of the relaxation time for the Cole-Cole relaxation. Additionally, this kind of relaxation is usually employed to investigate the ionic conductivity in condensed matter.

### 3.1.1. Theoretical consideration for modeling the complex impedance

Based on the above consideration, we tried to further investigate the complex impedance (Z*) from the theoretical point of view as well as the validity of the equivalent circuit shown in Fig. 2(g). This circuit was proposed by Zhang Q et al. [12] to fit the electrical response of the GO-based supercapacitor. This equivalent circuit includes a resistance (Re) in series with three blocks composed of resistance in parallel with constant phase element: (R1//CPE1); (R2//CPE2) and (R3//CPE3). This model provided an initial and appropriate representation of the electrical behavior of the main interfaces. The electrochemical performance is the collective result of all three parts, which were assigned to the rGO surface, the GO/rGO interface, and the GO interlayer respectively.

The deconvolution procedure was performed in order to isolate the three relaxation processes at low (LF); medium (MF) and high (HF) frequencies corresponding to the rGO surface, the GO/rGO interface, and the GO interlayer respectively. This can be achieved only by selecting the most appropriate sub-circuit corresponding to each relaxation behavior as shown in Fig. 4.

In our case, the subcircuit (Fig. 5) was selected) as a typical example to analyze the expression of its own complex impedance (Z*) data from theoretical point of view. It is assigned to the contribution of the rGO/rGO interlayer corresponding to the high-frequency relaxation (HF).

The constant phase element (CPE-T, p) elements are used in the above subcircuit instead of a capacitor taking into account, the distribution of relaxation times due to surface inhomogeneities [26]. The complex impedance of the constant phase element, (CPE-T, p) is given as:

$$ Z_{CPE} = \frac{1}{T(j\omega)^p} \quad (2) $$

where T is the pseudo-capacitance; ω is the angular frequency; j² = -1 is the imaginary number, and p is the (CPE) exponent.

The complex impedance Z*(ω) for such a subcircuit can be written as follows:

$$ Z^*(\omega) = R_e + \frac{R_1}{1 + (j\omega\tau_{z1})^{p1}} \quad (3) $$

Where (τz) is denoting the relaxation time:

$$ \tau_{z1} = (R_1 T_1)^{1/p1} \quad (4) $$

<figure><img src="image_7.png" /><figcaption>Fig. 5. Subcircuit corresponding to the Cole-Cole relaxation at high frequency relaxation (HF).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 4. Equivalent circuit used to analyze the complex impedance spectra (Z*).</figcaption></figure>
Therefore, the expression of the complex impedance ($Z^*$) is found to be similar to that shown before in the expression (1) to describe the Cole-Cole relaxation for the complex permittivity ($\epsilon^*$).

The decomposition of the complex impedance ($Z^*$) into its real and imaginary parts results in the following expressions:

$$Z(\omega) = R_e + \frac{R_1 [1 + \tau_{z1}^{p_1} \omega^{p_1} \cos(p_1 \times \pi/2)]}{(1 + \tau_{z1}^{p_1} \omega^{p_1} \cos(p_1 \times \pi/2))^2 + (\tau_{z1}^{p_1} \omega^{p_1} \sin(p_1 \times \pi/2))^2} \quad (5)$$

$$Z''(\omega) = \frac{R_1 \tau_{z1}^{p_1} \omega^{p_1} \sin(p_1 \times \pi/2)}{(1 + \tau_{z1}^{p_1} \omega^{p_1} \cos(p_1 \times \pi/2))^2 + (\tau_{z1}^{p_1} \omega^{p_1} \sin(p_1 \times \pi/2))^2} \quad (6)$$

From the theoretical point of view; the critical angular frequency corresponding to the peak can be determined from the derivative of the imaginary part of $Z''(\omega)$ with respect to angular frequency and by solving the following equation: $dZ''(\omega)/d\omega = 0$.

In this case, the angular frequency corresponding to the maximum for the imaginary part of $Z''(\omega)$ was found to be as follows:

$$\omega_{\max} = \frac{1}{\tau_{z1}} = \frac{1}{(R_1 \cdot T_1)^{1/p_1}} \quad (7)$$

As a result, the complex impedance $Z^*(\omega)$ can be described by the following expression similar to that representing the Cole-Cole relaxation process [27].

$$Z^*(\omega) = Z_{\infty} + \frac{Z_s - Z_{\infty}}{1 + (j\omega\tau_z)^p} \quad (8)$$

Where $\tau_z=(2\pi f)^{-1} = 1/\omega_{max}$ represents the relaxation time, $\omega_{max}$ is the maximum angular frequency, $\Delta Z = Z_s-Z_\infty$ is the electrical strength, ($Z_\infty = R_e$) corresponds to the impedance value at high frequency, and ($Z_s = R_e + R_1$) corresponds to the impedance value at low frequency.

The global impedance ($Z^*$) of the equivalent circuit given in Fig. 4 can be represented as the summation of the impedance of sub-circuit corresponding to each block:

$$Z^*(\omega) = R_e + Z_{rGO/rGO}(\omega) + Z_{rGO/GO}(\omega) + Z_{GO/GO}(\omega) \quad (9)$$

$$Z^*(\omega) = R_e + \frac{R_{rGO}}{1 + (j\omega\tau_{z,rGO})^{p_{rGO}}} + \frac{R_{interface}}{1 + (j\omega\tau_{z,interface})^{p_{interface}}} + \frac{R_{GO}}{1 + (j\omega\tau_{GO})^{p_{GO}}} \quad (10)$$

To keep consistency with the same notation used in the previous subcircuit (Fig. 5), we can rewrite:

$$Z^*(\omega) = R_e + \frac{R_1}{1 + (j\omega\tau_{z,1})^{p_1}} + \frac{R_2}{1 + (j\omega\tau_{z,2})^{p_2}} + \frac{R_3}{1 + (j\omega\tau_{z,3})^{p_3}} \quad (11)$$

Where the three expressions of the three relaxation times corresponding to each Cole-Cole relaxation process are:

$$\tau_{z1} = (R_1 \cdot T_1)^{1/p_1}, \quad \tau_{z2} = (R_2 \cdot T_2)^{1/p_2} \text{ and } \tau_{z3} = (R_3 \cdot T_3)^{1/p_3} \quad (12)$$

Therefore, the evolution of the global complex impedance ($Z^*$) can be investigated by the superposition of three the Cole-Cole relaxation process.

### 3.1.2. Deconvolution approach of the different relaxations process

The deconvolution procedure of the various processes such as relaxation and diffusion could offer the possibility to isolate and to identify valuable information insight into the individual contribution of each process. Therefore, a deconvolution was performed in order to isolate the three relaxation processes at low (LF); medium (MF), and high (HF) frequency as well as to validate the above equivalent circuit. As it was indicated before, this procedure can be achieved only by the

selection of the appropriate sub-circuit involved in the equivalent circuit.

Fig. 6(a and b) shows the deconvolution procedure of the different relaxation processes involved in the evolution of the real ($Z'$) and imaginary part ($Z''$) versus frequency in the range $[10^{-3}-3.10^6 \text{ Hz}]$.

As illustrated in Fig. 6(a and b), the first maximum is observed in the evolution of the imaginary part $Z''(F)$ indicating the relaxation process at low frequency (LF) around $8,44 \cdot 10^{-2}$ Hz corresponding to the graphene oxide (GO/GO) interlayers. The second maximum is observed in the intermediate frequency (MF) region around $4,35 \cdot 10^{-1}$ Hz for the relaxation process attributed to the (rGO/GO) interface. The third relaxation process appeared at a high frequency (HF) around $1,67 \cdot 10^5$ Hz for the (rGO/rGO) interlayer. Moreover, It is important to note that the evolution of the real part ($Z'$) of the complex impedance ($Z^*$) versus frequency exhibited simultaneously a significant change in the slope for each value of the frequency corresponding to the observed maxima in the evolution of the imaginary part $Z''(F)$.

Therefore, the deconvolution procedure highlights and confirms the existence of three Cole-Cole relaxation processes. These relaxation processes correspond to three distinct interfaces, as indicated in the circuit provided by Zhang. Q et al. [12]: The graphene oxide (GO); (rGO/GO) interface and the reduced Graphene (rGO) at low (LF); medium (MF) and high frequency (HF) respectively.

The obtained values from our deconvolution procedure and theoretical consideration of the relaxation times as well as their corresponding frequencies at low; medium and high frequencies respectively are reported in Table 2.

As shown from the above Tables 2 and it is found that these values are very close indicating that our deconvolution procedure and theoretical consideration were very useful in identifying the three relaxations process corresponding to interfaces GO/GO, rGO/GO, and rGO/rGO. More interestingly, both approaches allowed us to confirm the existence of the second maximum in the medium-frequency (MF) region, which was hidden by the maximum in the low-frequency region.

Therefore, the deconvolution procedure carried out on the analysis of the complex impedance ($Z^*$) data highlights and confirms the existence of three Cole-Cole relaxation processes as well as the validity of the equivalent circuit proposed by Zhang. Q et al. [12].

### 3.2. Analyses of the complex conductivity $\sigma^*(\omega)$

Electrical conductivity, denoted as $\sigma^*(\omega)$, is a measure of the material's ability to carry electrical current. It describes how easily an electric charge can flow through the material when subjected to an alternating electric field. The data of complex conductivity $\sigma^*(\omega)$ can be derived from the complex conductivity data following the relationship:

$$\sigma^*(\omega) = k \frac{1}{Z^*(\omega)} \quad (13)$$

$$\sigma^*(\omega) = \frac{k}{|Z^*(\omega)|^2} (Z(\omega) + jZ''(\omega)) \quad (14)$$

where k is the constant cell and $Z^*(\omega)$ is the complex impedance.

The real part $\sigma'(\omega)$ and the imaginary part $\sigma''(\omega)$, are related to the angular frequency $\omega$ by the following equations:

$$\sigma'(\omega) = k \frac{Z'(\omega)}{|Z'(\omega)|^2} \quad (15)$$

$$\sigma''(\omega) = k \frac{Z''(\omega)}{|Z''(\omega)|^2} \quad (16)$$

It has been pointed out [12], that in the supercapacitor sandwich structure (rGO/GO/rGO), the graphene oxide exhibits relatively lower conductivity compared to reduced graphene by effectively trapping water and ions within the interspaces of GO films due to functional
<figure><img src="image_9.png" /><figcaption>Fig. 6. (a–b). Deconvolution of the different relaxation processes in the evolution of the real (Z') and imaginary part (Z'') versus frequency.</figcaption></figure>

<table><caption>**Table 2**
Relaxation time and frequency for the three interfaces GO/GO, rGO/GO, and rGO/rGO.</caption><thead><tr><th colspan="3">High-frequency process rGO/rGO</th><th colspan="3">Medium frequency process rGO/GO</th><th colspan="3">Low-frequency process GO/GO</th></tr><tr><th>τz (s)</th><th>f th(Hz)</th><th>f simu (Hz)</th><th>τz (s)</th><th>f th (Hz)</th><th>f simu(Hz)</th><th>τz (s)</th><th>f th (Hz)</th><th>f simu(Hz)</th></tr></thead><tbody><tr><td>9,53.10<sup>-7</sup></td><td>1,67.10<sup>5</sup></td><td>1,71.10<sup>5</sup></td><td>3,66.10<sup>-1</sup></td><td>4,35.10<sup>-1</sup></td><td>4,17.10<sup>-1</sup></td><td>1,88</td><td>8,44.10<sup>-2</sup></td><td>7,93.10<sup>-2</sup></td></tr></tbody></table>

groups (hydroxyl, carboxyl, and epoxide groups) onto its surface. In contrast, the reduced graphene oxide (rGO) exhibits an enhanced electrical conductivity due to the reduction of the oxygenated functional groups present on the graphene oxide's surface. Additionally, it is stated by Zhang. Q et al. [12] that the functional groups and the water molecules, which come from the preparation of the GO samples are crucial for obtaining a considerable capacitance of the sandwiched GO capacitors. These later can be considered as the two prerequisites for forming charge separation under charging: Indeed, functional groups provide ions (such as protons from hydrolysis of the functional groups), and water molecules serve as carriers for the migration of ions either via its hydrogen-bonding network or in the free H3O⁺ form; the above super-capacitor sandwich structured (rGO/GO/rGO).

Given the above statement again, it is evident the investigation of the complex conductivity is of great importance. Therefore; it seems to us very important to explore and to investigate the behavior of the complex conductivity (σ*) in the current study.

Fig. 7(a and b) shows the evolution of the real part (σ') and the imaginary part (σ'') of the complex conductivity (σ*) in the same frequency range [10⁻³–3.10⁶ Hz] explored for the complex conductivity (Z*).

It can be seen in the Nyquist plot Fig. 7 (a) that the evolution of the

imaginary part (σ') versus the real part (σ') of the complex impedance (σ*) of the supercapacitors (rGO/GO/rGO) showed small asymmetric depressed semicircle in the medium and low-frequency region suggesting the existence of tow depressed semicircle. At high frequencies, a highly depressed semicircle seems incomplete.

Additionally; there is a significant difference in the diameter of these semicircles. This could explained by the values of the semicircle radius observed in the Nyquist plot for the complex impedance (Z*). These later serve as an indicator of conductivity, which is related to the inverse of the charge-transfer resistance (R<sub>CT</sub>) that governs the limiting process at the rGO/rGO, rGO/GO, or GO/GO interfaces [25]. It is expected that an opposite trend becomes apparent when examining the RCT values in the Nyquist plots as well as from the value obtained with the equivalent circuit in Table 1: R<sub>CT</sub>(GO/GO) > R<sub>CT</sub>(rGO/GO) > R<sub>CT</sub>(rGO/rGO). Consequently, this demonstrates that the semicircle radius observed in the Nyquist plots for the complex impedance (σ*).

It can be seen from Bode plots (Fig. 7(b)) that the imaginary part (σ') as a function of frequency, revealed the existence of two distinct regions in the frequency range as observed in the imaginary part (Z'') for the complex impedance (Z*). However, only one very broad maxima and asymmetric curve can be observed and are located in the low (LF) and medium (MF) frequency region. This behavior could suggest the

<figure><img src="image_10.png" /><figcaption>Fig. 7. (a–b). Complex conductivity (σ*) in the frequency range [10⁻³–3.10⁶ Hz]. Nyquist (a) and Bode (b) plots.</figcaption></figure>
existence of overlapping maximum in the low and medium frequency regions.

In contrast, in the high-frequency region, it can be seen that the imaginary part (σ') versus frequency continues to increase above the high limit frequency (3.10⁶ Hz).

In addition, from the evolution of the part (σ') as a function of frequency, it can be also seen that there is a change in the slopes in the low (LF) and medium (MF) frequency regions as well as in the high frequency (HF) region.

Consequently, these behaviors lead us to wonder about the existence of a third maximum that could appear above (3.10⁶ Hz) in the evolution of the imaginary part (σ') as a function of frequency. Therefore, an exploration approach was performed up to 3.10⁹ Hz based on this finding.

### 3.2.1. Extrapolation approach for conductivity spectra

Fig. 8(a and b) shows a superposition of the evolutions of the imaginary part (Z') and (σ') of the complex impedance (Z*) and conductivity (σ*) in the frequency range [10⁻³–3.10⁶ Hz] as well the extrapolation up to (3.10⁹ Hz).

Fig. 8(a) shows the existence of two distinct regions: For the first region both evolutions exhibited overlapping maxima in the low and medium frequency region. In contrast, for the second region, a very clear maximum can be seen only in the evolution of the imaginary part (Z') versus frequency. After extrapolation up to (3.10⁹ Hz), a very clear maximum appeared also in the evolution of the imaginary part (σ') versus frequency in the second region. Moreover, it is found that the frequencies corresponding to the maxima observed in the evolution of the imaginary part (σ') versus frequency are located above those in the evolution of the imaginary part (Z') versus frequency.

This finding strongly suggests the occurrence of dynamic behavior similar to the relaxation phenomenon. Upon further analysis of the complex conductivity (σ*) spectra in both Nyquist and Bode plots, it becomes evident that three well-defined and separate peaks at specific frequencies [22].

In addition, the evolutions of complex conductivity (σ*) in Bode and Nyquist plots reflect the same behavior and the evolutions for the complex permittivity (ε*), which is usually used to describe the behavior of the Cole-Cole relaxation [25].

As a result, the next step will involve further investigation of the complex conductivity (σ*) as it was conducted for the complex impedance (Z*) from a theoretical point of view. The aim is to analyze all three observed relaxation processes and to determine their origin as well as the validity of the equivalent circuit.

### 3.2.2. Theoretical consideration for modeling the complex conductivity

The subcircuit (Fig. 5) selected to analyze the expression of its complex impedance (Z*) from the theoretical point of view was also used for developing the theoretical expression of the complex

conductivity (σ*).

This subcircuit was assigned to the contribution of the (rGO/rGO) interlayer corresponding to the high frequency (HF) relaxation process that was described by the Cole-Cole relaxation. The evolutions of the imaginary part (Z') and (σ') versus frequency are similar. It was found that the frequency corresponding to the maximum observed in the evolution of the imaginary part (σ') versus frequency is located above that for the evolution of the imaginary part (σ') versus frequency.

As a result, the complex conductivity σ*(ω) is expected to be described by the Cole-Cole relaxation process [27]:

$$ \sigma * (\omega) = \sigma_{\infty} + \frac{\sigma_s - \sigma_{\infty}}{1 + (i\omega\tau_{\sigma})^p} \quad (17) $$

where, (σ∞) corresponds to the conductivity value at high frequencies, (Δσ = σs-σ∞) is the ionic strength, (σs) corresponds to the conductivity value at low frequencies and τσ is the relaxation time.

It is related to the angular frequency (ωmax) corresponding to the maximum angular frequency:

$$ \tau_{\sigma} = 1/\omega_{\max} = (2\pi f)^{-1} \quad (18) $$

As it was mentioned before, the complex conductivity (σ*) is related to the complex impedance (Z*) in the following relationship:

$$ \sigma^*(\omega) = k \times \frac{1}{Z^*(\omega)} \quad (19) $$

It was found that the frequency corresponding to the maximum observed in the evolution of the imaginary part (σ') versus frequency is located above to those in the evolution of the imaginary part (Z') versus frequency.

The expressions of the real part σ'(ω) and the imaginary part σ''(ω) through the real part Z'(ω) and the imaginary part Z''(ω):

$$ \sigma'(\omega) = \frac{(R_e + R_1) + (2R_e + R_1)\tau_{z1}^{p1}\omega^{p1} \cos(p_1 \times \pi/2) + R_e\tau_{z1}^{2p1}\omega^{2p1}}{(R_e + R_1)^2 + 2(R_e + R_1)R_e\tau_{z1}^{p1}\omega^{p1} \cos(p_1 \times \pi/2) + R_e^{2p1}\tau_{z1}^{2p1}\omega^{2p1}} \quad (20) $$

$$ \sigma''(\omega) = \frac{R_1 \tau_{z1}^{p1} \omega^{p1} \sin(p_1 \times \pi/2)}{(R_e + R_1)^2 + 2(R_e + R_1)R_e \tau_{z1}^{p1} \omega^{p1} \cos(p_1 \times \pi/2) + R_e^{2p1} \tau_{z1}^{2p1} \omega^{2p1}} \quad (21) $$

Similarly, by using the derivative of the expression of the imaginary part of σ''(ω) with respect to angular frequency and solving the equation (d(σ''(ω)/dω) = 0), the expression of the relaxation time (τσ = 1/ωmax = 2πf):

$$ \tau_{\sigma} = \left( \frac{R_e}{R_e + R_1} \right)^{1/p_1} \cdot \tau_z \quad (22) $$

<figure><img src="image_11.png" /><figcaption>Fig. 8. (a–b). (a). Superposition of the imaginary part (Z') and (σ') of the complex impedance (Z*) and conductivity (σ*) in the frequency range [10⁻³–3.10⁶ Hz]. (b). Extrapolation [10⁻³–3.10⁹ Hz].</figcaption></figure>
On the other hand, the expressions of the real part $\sigma'(\omega)$ and the imaginary part $\sigma''(\omega)$ can be obtained from the Cole-Cole relaxation expression (15).

$$ \sigma'(\omega) = \frac{(\sigma_{\infty} + \Delta\sigma) + (2\sigma_{\infty} + \Delta\sigma)\tau_{\sigma}^{p1} \cos(p_1 \times \pi/2) \times \omega^{p1} + \sigma_{\infty}\tau_{\sigma}^{2p1} \times \omega^{2p1}}{[1 + 2\tau_{\sigma}^{p1} \cos(p_1 \times \pi/2) \times \omega^{p1} + \tau_{\sigma}^{2p1} \times \omega^{2p1}]} \quad (21) $$

$$ \sigma''(\omega) = \frac{\Delta\sigma\tau_{\sigma}^{p1} \sin(p_1 \times \pi/2) \times \omega^{p1}}{[1 + 2\tau_{\sigma}^{2p1} \cos(p_1 \times \pi/2) \times \omega^{p1} + \tau_{\sigma}^{2p1} \times \omega^{2p1}]} \quad (22) $$

The identification between both expressions of $\sigma''(\omega)$, obtained from $Z''(\omega)$ and that obtained from the expression of Cole-Cole relaxation, lead us to extract the all expressions involved in the Cole-Cole relaxation process such as the conductivity ($\sigma_s$) and ($\sigma_{\infty}$) at low and high frequencies as well as the ionic strength ($\Delta\sigma = \sigma_s - \sigma_{\infty}$).

Therefore, the main parameters describing one relaxation can be expressed as follows:

$$ \sigma_s = k \frac{1}{R_e + R_1} \quad (21); \quad \sigma_{\infty} = \frac{k}{R_e} \quad \text{and} \quad (22) \quad \tau_{\sigma} = \left( \frac{R_e}{R_e + R_1} \right)^{1/p_1} \cdot \tau_z \quad (23) $$

Given that, theoretical examination of the global complex conductivity can be expressed as the sum of three analogous Cole-Cole relaxations.

$$ \sigma^*(\omega) = \sigma_{\infty} + \frac{\Delta\sigma_{LF}}{1 + (i\omega\tau_{\sigma LF})^{p1}} + \frac{\Delta\sigma_{MF}}{1 + (i\omega\tau_{\sigma MF})^{p2}} + \frac{\Delta\sigma_{HF}}{1 + (i\omega\tau_{\sigma HF})^{p3}} \quad (24) $$

For a more comprehensive analysis and visualization of the three relaxations occurring at low, medium, and high frequencies, the deconvolution approach will be employed. This approach allows us to distinguish and isolate the individual contributions of each relaxation within the extrapolated frequency range $[10^{-3}-3.10^9 \text{ Hz}]$.

### 3.2.3. Deconvolution approach for the conductivity spectra

The spectral deconvolution process facilitates the discernment of three distinct relaxations that collectively constitute the overall conductivity, as shown in Fig. 9(a and b) in the Nyquist and Bode representations.

The Nyquist plot, as depicted in Figure (9-a), has been analyzed to determine the electrical behavior of the three relaxations present in different layers of the supercapacitor. The imaginary conductivity ($\sigma''$) versus the real conductivity ($\sigma'$) were recorded over a wide frequency range $[10^{-3}-3.10^9 \text{ Hz}]$, revealing the formation of three semi-circular arcs indicative of three Cole-Cole type relaxations [27]. The inset in Figure (8-a) highlights the deconvolution of the two low (LF) and medium (MF) frequency relaxation processes, which are not well discernible in the Nyquist plot as they were merging and dominated by the frequency relaxation.

Fig. 9 (b) showed very clear three relaxation peaks from the

evolution of the imaginary part ($\sigma''$) versus frequency in the two distinct regions: In the first region of the imaginary part ($\sigma''$) versus frequency evolutions exhibited an overlapping maximum in the low and medium frequency region. After extrapolation up to (3.109 Hz), a very clear maximum appeared also in the evolution of the imaginary part ($\sigma''$) versus frequency in the second region.

The first peak occurs at a low frequency of approximately $1,93.10^{-1}$ Hz, which is associated with the interlayer graphene oxide (GO) due to the presence of the functional groups (-OH, -COOH, and C=O) as well as two types of water molecules, namely mobile water and confined water within the GO films [12], facilitating interfacial polarization. The second peak, at 1.89 Hz, is attributed to the rGO/GO interface. The final peak, at a high frequency of $1.8 \ 10^7$ Hz, corresponds to the rGO interlayers.

The first relaxation occurs at low frequencies where charge accumulation takes place due to charged surfaces of the GO interlayer, which tend to attract oppositely charged species, resulting in a decrease in conductivity. The second relaxation occurs at the intermediate rGO/GO interface, and the last conductivity dispersion occurs at high frequencies, where conductivity is significantly enhanced due to delocalized charge carriers that contribute to long-range conduction, particularly within the rGO interlayer.

As was shown for the analysis of the complex impedance ($Z^*$), the obtained values from our deconvolution procedure and theoretical consideration of the relaxation times as well as their corresponding frequencies are reported in Table 3 for a comparison.

It can be seen that the values obtained from our deconvolution procedure and theoretical consideration in addition to the extrapolation are very close. Moreover, these results allowed us to confirm the existence of the third relaxation process in the high frequency(MF) region above 3 MHz. In addition, the separation of the overlapping relaxations in the low and medium frequency regions.

The deconvolution procedure and theoretical consideration in addition to the extrapolation were very useful in identifying the three relaxation processes corresponding to interfaces: GO/GO, rGO/GO, and rGO/rGO.

### 3.3. Correlation between the electrical properties

Based on the deconvolution procedure, allowed the identification of the three Cole-Cole relaxation processes involved in the evolution of the global complex conductivity. This finding is of great importance in describing the dynamic behavior, which can occur in these three parts of the supercapacitor. Indeed, the main electrical properties corresponding to the three specific parts (rGO/rGO, rGO/GO, and GO/GO) were extracted such as The relaxation time ($\tau_e$), ionic strength ($\Delta\sigma$), high-frequency conductivity ($\sigma_{\infty}$), and low-frequency conductivity ($\sigma_s$).

These parameters obtained from the low (LF), medium (MF), and high frequency (HF) relaxation processes are plotted in the histogram for

<figure><img src="image_12.png" /><figcaption>Fig. 9. (a–b): Deconvolution of the complex conductivity of the sandwiched GO films in the extrapolated frequency range $[10^{-3}-3.10^9 \text{ Hz}]$. Nyquist (a) and Bode (b).</figcaption></figure>
<table><caption>**Table 3**
Relaxation time and frequency for the three interfaces GO/GO, rGO/GO, and rGO/rGO.</caption><thead><tr><th colspan="3">High-frequency process rGO/rGO</th><th colspan="3">Medium frequency process rGO/GO</th><th colspan="3">Low-frequency process GO/GO</th></tr><tr><th>τσ (s)</th><th>f th(Hz)</th><th>fsimu (Hz)</th><th>τσ (s)</th><th>f th (Hz)</th><th>fsimu (Hz)</th><th>τσ (s)</th><th>f th (Hz)</th><th>fsimu (Hz)</th></tr></thead><tbody><tr><td>8,82.10<sup>-9</sup></td><td>1,80.10<sup>7</sup></td><td>1,70.10<sup>7</sup></td><td>8,42.10<sup>-2</sup></td><td>1,89</td><td>1,77</td><td>8,26.10<sup>-1</sup></td><td>1,93.10<sup>-1</sup></td><td>1,76.10<sup>-1</sup></td></tr></tbody></table>

a comparison between them as shown in Fig. 10(a and b).

It can be observed from Fig. 10 (a) that the relaxation time $τ_σ$ decreases from GO to rGO layers whereas the ionic strength ($Δσ$) increases. These parameters could describe the dynamic behavior of three specific regions: (rGO/rGO, rGO/GO, and GO/GO).

Taking the example of the GO/GO interlayers: The presence of oxygen groups, mainly hydroxyl, carbonyl, and carboxyl, as well as under an external electric field, the $H_3O^+$ protons will reorient and lodge themselves in the interlayer spaces, confining them to a restricted space, thus limiting their mobility [28]. This leads to localized movement and a more compact structure resulting in dynamic behavior with a long timescale that can be observed in the low frequency (LF) [29]. This behavior correlates very well with the low values obtained for the conductivity ($σ_s$) and ($σ_∞$) at low and high frequencies as shown in Fig. 10 (b).

Conversely, for the reduced graphene oxide (rGO) layer. It is known that after reduction, a significant portion of functional groups is lost, resulting in reduced attraction of water molecules with less compact structure indicating a dynamic behavior with a short timescale that can be observed in the high frequency (HF). As a result, the mobility of ions is more favorable, leading to an increase of the values for the conductivity ($σ_s$) and ($σ_∞$) at low and high frequency as it also illustrated from Fig. 10 (b).

At the rGO/GO interface, due to the existence of both layers, an intermediate behavior is observed. Hence, the rank for the conductivity ($σ_s(rGO) > σ_s(rGO/GO) > σ_s(GO)$) was obtained due to reduction and the presence of water molecules in the interlayer spacing. In contrast, the three interface components exhibit three diverse and uniformly distributed relaxations in the frequency range $[10^{-3}-3.10^9$ Hz] with the inverse rank: $τ_σ(GO) > τ_σ(rGO/GO) > τ_σ(rGO)$.

The following relationship between the relaxation time and the diffusion coefficient was also employed to further support the above analysis used by Refs. [30,31].

$$τ_∞ = \frac{L}{D} \qquad (25)$$

where $τ$ is the relaxation time, $D$ is the diffusion coefficient, and $L$ denotes the length scale.

The low frequency (LF) relaxation process observed in GO/GO interlayers indicates a high relaxation time $τ_σ(GO)$ due to a more compact structure resulting in a slow process and therefore a low diffusion coefficient is expected to be obtained. This finding is a good agreement with the low values for the low and high-frequency conductivity ($σ_s$) and ($σ_∞$).

For the high-frequency (HF) relaxation process in rGO/rGO interlayers, the low value of the relaxation time $τ_σ(rGO)$ indicates a fast process resulting in a high diffusion coefficient. Consequently, high values are obtained for the low and high-frequency conductivity ($σ_s$) and ($σ_∞$).

For the intermediate frequency (MF) relaxation process at the rGO/GO interface, which exhibited a relaxation time $τ_σ(GO)$ value between the two above relaxation processes. This indicates a moderate process leading to a moderate diffusion coefficient resulting in the values for the low and high-frequency conductivity ($σ_s$) and ($σ_∞$) between the two above relaxation processes.

In order to support this finding, we extracted the equivalent capacitance ($C_{equi}$) for the three specific regions (GO/GO), rGO/GO, and rGO/rGO) from the relaxation time values corresponding to the three relaxation processes at low, medium, and high frequencies.

The equivalent capacitance ($C_{eq}$) can be determined from the following relationship:

$$C_{eq} = \left( \frac{R_e + R_1}{R_e R_1} \right) τ_σ \qquad (26)$$

The obtained values as well as the low and high frequency conductivity ($σ_s$) and ($σ_∞$) are plotted in the histogram for a comparison between them as shown in Fig. 11(a and b).

It is also found that the equivalent capacitance ($C_{eq}$) is ranked as follows ($C_{eq}(GO) > C_{eq}(rGO/GO) > C_{eq}(rGO)$) indicating that the GO/GO interlayers exhibit a high value. This correlate also very well with a new model, namely the “Charge Close-Packed model” (CCP) proposed by Zhang. Q et al. [12] explain the abnormal capacitive behavior of the sandwiched GO film. In the (CCP) model, extensive numbers of layers of alternatively packed positive and negative charges are periodically ordered one by one, with no diffusion regions present. Because of the confinement of the GO and (rGO) layers together with the poor ionic mobility, the diffusion length of the ions has been restricted within the nanoscaled interlayer spacing.

Hydrated graphene oxide (GO) films can enhance dielectric strength through the intercalation of water molecules, offering capacitances per unit area of 100–800 mF/cm², 5 to 40 times higher than traditional activated carbon materials. The performance of these dielectric capacitors depends on water content and applied voltage, illustrating their potential for next-generation energy storage devices [32].

Therefore, the rank for the conductivity ($σ_s(GO) < σ_s(rGO/GO) < σ_s(GO)$) could illustrate the obtained results due to reduction and the

<figure><img src="image_13.png" /><figcaption>**Fig. 10.** (a–b). (a) Relaxation time ($τ_σ$) and ionic strength ($Δσ$). (b) Low ($σ_s$) and high-frequency conductivity ($σ_∞$) at the three specific regions: (rGO/rGO, rGO/GO, and GO/GO).</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. 11.** (a–b). (a) Equivalent capacitance (C<sub>eq</sub>) and low-frequency conductivity (σ<sub>s</sub>). (b) Equivalent capacitance (C<sub>eq</sub>) and high-frequency conductivity (σ<sub>∞</sub>) at the three specific regions: (GO/GO, rGO/GO, and rGO/rGO).</figcaption></figure>

presence of water molecules in the interlayer spacing [27]. This behavior correlates very well once again with the phenomenon that leads to the creation of a dipole moment, which is responsible for the increase in capacitive behavior.

## 4. Conclusion

The storage mechanism of graphene oxide-based solid-solid supercapacitors structured as (rGO/GO/rGO) was investigated through the analysis of the complex impedance (Z*) combined with the complex conductivity (σ*) by using simultaneously Bode and Nyquist diagram. This approach can be considered the most appropriate approach to investigate the different processes involved in this supercapacitor structured as rGO/GO/rGO. It is also very useful to support and confirm the results based on the density functional theory (DFT) method to explain the anomalous capacitive behavior of the GO sandwich film by highlighting alternately positively and negatively charged layers without diffusion zones.

From the analysis of the complex impedance (Z*) in the frequency range from 3.10⁶ Hz, it is found that our theoretical approach as well as the deconvolution procedure were very useful in confirming the existence of three distinct Cole relaxation processes involved in the storage and conduction mechanism in this supercapacitor. The three relaxation processes were attributed to the rGO/rGO, the rGO/GO, and the GO/GO interfaces.

Moreover; further analysis of the complex conductivity (σ*) based on the extrapolation and the deconvolution procedures in the frequency range up to 3.10⁹ Hz. Both approaches allowed us to further investigate and highlight the existence of third relaxation processes above 3.10⁶ Hz as it was observed in the complex impedance (Z*) as well as the conductivity (σ*). According to our theoretical approach carried out on they are also found to be similar to the Cole-Cole relaxation corresponding to the Maxwell-Wagner-Sillars (MWS) relaxation due to the local electrostatic interactions between the active sites on molecular chains and the counter ions leading to the creation of induced dipolar moments or diffusion process. The relaxations at low, medium, and high frequency in correlation with (the DFT) result are attributed to three specific regions respectively the graphene oxide (GO); (rGO/GO) interface, and the reduced Graphene (rGO). A good correlation between the electrical properties and the charge conduction and storage mechanisms mechanism was obtained.

Our approach based on a deep analysis of the conductivity (σ*) was very useful in highlighting the charge conduction and storage mechanisms in (rGO/GO/rGO) supercapacitor films.

## Funding

This research received no external funding.

## CRediT authorship contribution statement

**Dalal Hilmi:** Writing – original draft, Investigation, Formal analysis, Conceptualization. **Soumia Zaim:** Writing – original draft, Formal analysis. **Abdehadi Mortadi:** Writing – original draft, Formal analysis. **Ibtissam Sabir:** Formal analysis. **Mohamed Monkade:** Supervision. **Rachid Nmila:** Supervision. **Alina Violeta Ursu:** Supervision. **Philippe Michaud:** Supervision. **Halima Rchid:** Supervision. **Reddad El Moznine:** Writing – review & editing, Validation, Supervision.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Data availability

No data was used for the research described in the article.

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