# Physicochemical Modeling of Electrochemical Impedance in Solid-State Supercapacitors

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## Abstract

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## 1. Introduction

## 2. Experimental Investigations

#### 2.1. Preparation of Biocarbon and Free-Standing bc−GP Electrodes

_{4}(Acros Organics, Geel, Belgium) solution (10 mL). Here, MgSO

_{4}was used as an activating agent to enlarge existing pores and allow the formation of new pores by opening the inaccessible pores, thereby preventing ash formation [40]. The reaction was hydrothermally treated for 12 h at 230 °C. The obtained mixture was dried for another 12 h in a hot-air oven at 60 °C. The dried powder mixture was calcined for 2 h in a tubular furnace at 700 °C and heated up with 10 °C/min ramp under an inert argon atmosphere. The as-obtained black carbon derived from biomass was used as an active electrode material, biologically based carbon (BioCarbon), hereinafter referred to as bc−GP. The bc−GP powder materials were made into free-standing electrodes using a dry-film process for electrochemical characterization. To fabricate these electrodes, a mixture of 0.2 g of Super P Carbon Black, 99.9+% (Alfa Aesar, Haverhill, MA, USA), and 1.8 g of bc−GP were ground manually using a mortar and pestle. This mixture was homogenized with 200 μL of PTFE solution (60 wt.% dispersion in water, Sigma-Aldrich, St. Louis, MO, USA) in a mortar and pestle containing 60 mL of ethanol. The above mixture was blended for 30 to 45 min until a soft dough-like mass was acquired. The dough was rolled into a sheet and dried for at least 12 h at 80 °C to obtain a 0.17 mm thick sheet, which was later cut into 10 mm disc-shaped electrodes.

#### 2.2. Preparation of Gel-Polymer Electrolyte

#### 2.3. Fabrication of SSC Cell

#### 2.4. Microstructure, Surface Morphology, and Electrical Characterization

## 3. Impedance Modeling

#### 3.1. Mathematical Approach to Modeling

#### 3.2. Physicochemical Approach to Modeling

#### 3.2.1. Porous Electrode Model

#### 3.2.2. Gel-Polymer Electrolyte Model

#### 3.3. Total Physicochemical Model

#### 3.4. Fitting of EIS Data

## 4. Results and Discussion

#### 4.1. Microscale Analysis of Electrode Surface Morphology

#### 4.2. Comparison of Developed Models and Experimental Data

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Cyclic Voltammogram

**Figure A1.**Cyclic voltammogram of the manufactured symmetric SSC in aqueous KOH and gel polymer electrolyte at a current density of 50 mA/g for 1 to 1000 cycles.

#### Appendix A.2. Porous Electrode and Gel-Polymer Electrolyte Parameters

Parameter | Unit | Value | Description | Reference |
---|---|---|---|---|

${L}_{\mathrm{e}}$ * | $\mathrm{m}$ | $3.60\times {10}^{-4}$ | thickness of the electrolyte | |

$A$ * | ${\mathrm{m}}^{2}$ | $7.85\times {10}^{-5}$ | surface area of the electrode plate | |

$\mathsf{{\rm X}}$ * | $\mathrm{m}$ | $7.00\times {10}^{-4}$ | full element thickness | |

$t$ * | $\mathrm{m}$ | $1.7\times {10}^{-4}$ | thickness of the electrode | |

$l$ * | $\mathrm{m}$ | $5\times {10}^{-4}$ | total electrode pore depth | |

$\chi $ * | - | $3.5\times {10}^{-1}$ | electrode porosity volume fraction | |

$\mathrm{a}$ | $\mathrm{m}$ | $3.00\times {10}^{-10}$ | effective ionic radius | |

${r}_{\mathrm{p}}$ * | $\mathrm{m}$ | $5.51\times {10}^{-10}$ | spherical pore radius | |

${D}_{+}$ | ${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ | $7.84\times {10}^{-13}$ | diffusivity of Potassium ions in the electrolyte | [95] |

${D}_{-}$ | ${\mathrm{m}}^{2}{\mathrm{s}}^{-1}$ | ${10}^{-16}$ | diffusivity of electrons in the electrolyte | [96] |

${\epsilon}_{0}$ | ${\mathrm{F\; m}}^{-1}$ | $8.85\times {10}^{-12}$ | vacuum permittivity | [97] |

${\epsilon}_{\mathrm{i}}$ | - | 30 | relative permittivity | [98] |

${\epsilon}_{\mathrm{r}}$ | ${\mathrm{F\; m}}^{-1}$ | $78.5\times {10}^{-12}$ | permittivity at the stern/diffuse layer interface | [99] |

${E}_{\mathrm{a}}$ | $\mathrm{eV}$ | $2.5$ | activation energy | [95] |

${\sigma}_{\mathrm{e}}$ | ${\mathrm{S\; m}}^{-1}$ | $4.7\times {10}^{-2}$ | ionic conductivity at 298.15 k in electrolyte | [100] |

${c}_{\mathrm{max}}$ * | ${\mathrm{mol\; m}}^{-3}$ | $2.1\times {10}^{3}$ | maximum concentration | |

${c}_{+}$ * | ${\mathrm{mol\; m}}^{-3}$ | $1.05\times {10}^{3}$ | initial mobile ions concentration in the electrolyte | |

${c}_{-}$ * | ${\mathrm{mol\; m}}^{-3}$ | $1.05\times {10}^{3}$ | initial mobile electron concentration in the electrolyte | |

${z}_{+}$ | - | $+1$ | ion valence | |

${z}_{-}$ | - | $-1$ | electron valence | |

${\sigma}_{\mathrm{c}}$ | ${\mathrm{S\; m}}^{-1}$ | $4.7\times {10}^{-2}$ | conductivity of the electrode matrix | [101] |

$\kappa $ | ${\mathrm{S\; m}}^{-1}$ | $6.0\times {10}^{-5}$ | conductivity of the electrolyte | [100] |

${i}_{0}$* | $\mathrm{A}{\mathrm{m}}^{-2}$ | $6.37\times {10}^{2}$ | current density | |

$T$ * | $\mathrm{K}$ | $293.15$ | ambient temperature | |

F | ${\mathrm{C\; mol}}^{-1}$ | $96,485$ | Faraday constant | [102] |

R | ${\mathrm{J\; mol}}^{-1}{\mathrm{K}}^{-1}$ | 8.314 | gas constant | [102] |

${K}_{\mathrm{B}}$ | ${\mathrm{J\; K}}^{-1}$ | $1.380\times {10}^{-23}$ | Boltzmann constant | [103] |

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**Figure 1.**(

**a**) Disc-shaped PVA-KOH-KCl gel-polymer electrolyte; (

**b**) original setup; (

**c**) schematic of the fabricated SSC setup.

**Figure 3.**Schematic of an SSC and the corresponding dimensions ${L}_{\mathrm{s}}$ of the space charge layers and ${L}_{\mathrm{e}}$ of the total electrolyte length (graphic inspired by [44]).

**Figure 6.**Schematic representation of the Gouy–Chapman–Stern model and the formation of the Stern and space charge layers.

**Figure 9.**EC model for gel-polymer electrolyte as developed in [44].

**Figure 11.**FE-SEM imaging of (

**a**) porous bc−GP electrode; (

**b**) under higher magnification; (

**c**,

**d**) same sample from a different view and under higher resolution.

**Figure 12.**Nyquist plots of the SSC and its modeling as shown for (

**a**) the entire frequency spectrum; (

**b**) the high-frequency range; (

**c**) the real impedance ${Z}^{\prime}$ gainst frequency $\omega $; (

**d**) the imaginary impedance $Z\u2033$ against frequency $\omega $.

Parameter | Unit | Value | Description |
---|---|---|---|

${V}_{\mathrm{p}}$ | ${\mathrm{cc}\mathrm{g}}^{-1}$ | $1.540\times {10}^{-1}$ | Pore volume |

${A}_{\mathrm{p}}$ | ${\mathrm{m}}^{2}{\mathrm{g}}^{-1}$ | $2.689\times {10}^{2}$ | Surface area |

${d}_{\mathrm{p}}$ | $\mathrm{m}$ | $1.1031\times {10}^{-11}$ | Pore width |

Parameter $\overrightarrow{\mathit{p}}$ | Unit | Analytical Value | Value of Best Fit |
---|---|---|---|

Electrode | |||

${\tilde{R}}_{+}$ | Ω m^{−1} | $2.961\times {10}^{6}$ | $6.890\times {10}^{-2}$ |

${R}_{\mathrm{e}}$ | Ω m^{−1} | $15.32$ | $5.49$ |

$l$ | m | $5\times {10}^{-4}$ | $24.20$ |

${C}_{\mathrm{dl}}$ | F m^{−1} | $4.060$ | $0.160$ |

${C}_{\mathrm{flat}}$ | F | $5.790\times {10}^{-11}$ | $2.91\times {10}^{-2}$ |

Electrolyte | |||

${C}_{\mathrm{e}}$ | F | $5.790\times {10}^{-11}$ | $2.10\times {10}^{-7}$ |

${C}_{\mathrm{s},+}$ | F | $7.780\times {10}^{-1}$ | $3.35\times {10}^{-6}$ |

${C}_{\mathrm{s},-}$ | F | $7.780\times {10}^{-1}$ | $2.39$ |

${R}_{\mathrm{s},+}$ | Ω | $4.250\times {10}^{-11}$ | $14.30$ |

${R}_{\mathrm{s},-}$ | Ω | $3.330\times {10}^{-7}$ | $6.81$ |

${R}_{+}$ | Ω | $1.458\times {10}^{3}$ | $1.58\times {10}^{-1}$ |

${R}_{-}$ | Ω | $1.434\times {10}^{7}$ | $8.45$ |

${C}^{\mathsf{\delta}}$ | F | $1.813\times {10}^{3}$ | $7.15\times {10}^{-3}$ |

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**MDPI and ACS Style**

Peyrow Hedayati, D.; Singh, G.; Kucher, M.; Keene, T.D.; Böhm, R.
Physicochemical Modeling of Electrochemical Impedance in Solid-State Supercapacitors. *Materials* **2023**, *16*, 1232.
https://doi.org/10.3390/ma16031232

**AMA Style**

Peyrow Hedayati D, Singh G, Kucher M, Keene TD, Böhm R.
Physicochemical Modeling of Electrochemical Impedance in Solid-State Supercapacitors. *Materials*. 2023; 16(3):1232.
https://doi.org/10.3390/ma16031232

**Chicago/Turabian Style**

Peyrow Hedayati, Davood, Gita Singh, Michael Kucher, Tony D. Keene, and Robert Böhm.
2023. "Physicochemical Modeling of Electrochemical Impedance in Solid-State Supercapacitors" *Materials* 16, no. 3: 1232.
https://doi.org/10.3390/ma16031232