# Simulation of the Electrochemical Response of Cobalt Hydroxide Electrodes for Energy Storage

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## Abstract

**:**

^{®}simulation was developed on a diffusion and kinetics basis, simulating the equations of Fick’s second law and Butler–Volmer, respectively, towards understanding the energy-storage mechanisms of cobalt hydroxide electrodes. The simulation was compared to a real cobalt hydroxide system, showing an accurate approximation to the experimentally obtained response and deviations possibly related to other physical/chemical processes influencing the involved species.

## 1. Introduction

^{®}software). The main goal is to predict and model the electrochemical response of cobalt hydroxide electrodes via cyclic voltammetry. Cobalt hydroxide was chosen for this study as it is an interesting electrode material, known by its pseudocapacitive properties, that has been widely used in energy-storage devices such as supercapacitors and batteries [3,4,5].

## 2. Simulation Setup

^{®}) were developed to observe the current and voltage values, as well as the shape of the cyclic voltammetry (CV) curves, to draw conclusions regarding the behavior of the system. A critical discussion will follow the simulated CV curves to discuss how well, or poorly, simple mathematical models correlate to reality. Therefore, a comparative study will be presented comparing simulated and experimentally obtained CV curves.

#### 2.1. Potential Sweep

_{min}and ψ

_{max}, define the potential window (Δψ) and the shape of a potential sweep can be observed by plotting it in a function of time (Figure 1). The total duration of the sweep (t

_{CV}) can be obtained with Δψ and ν by using Equation (1). The potential throughout time can then be defined in two domains, as calculated, and presented in Equation (2).

#### 2.2. Butler–Volmer Kinetics

_{F}and j

_{0}are the faradaic current density and the exchange current density (A/m

^{2}), respectively; α is the system’s transfer coefficient; z is the number of transferred electrons; R is the gas law constant (J/K mol); T is the temperature (K); and η is the overpotential (V), which can be defined by Equation (4).

_{0}is the equilibrium potential of the system and ψ(t) is the potential sweep previously defined by Equation (2). There are several inherent parameters inside the Butler–Volmer equation that will vary during the cathodic and anodic reactions of the electrode. These variables will change depending on the kinetic order of the reactions occurring at the electrode active material. Therefore, for the cobalt hydroxide system, a fast and reversible redox reaction (Reaction I) will be the ruling mechanism of the faradaic response.

^{−}, and the reductant is water, while the electrode’s surface will change its composition from $\mathrm{Co}{\left(\mathrm{OH}\right)}_{2}$ to $\mathrm{CoOOH}$ during oxidation, and vice versa during reduction. Few further assumptions were made in order to simplify the problem at hand: no solid-state diffusion occurs in the bulk of the active material; the reaction will follow a heterogeneous zero-order (i.e., catalytic behavior); and there is no secondary reaction or accumulation of species occurring in the system.

#### 2.3. Diffusion Gradients

_{ox}and D

_{red}are the diffusion coefficients of the oxidant and the reductant, respectively; C

_{ox}and C

_{red}are the concentration of oxidant and reductant, respectively; and x is the distance of the redox species to the electrode’s surface. This differential equation is well-known in chemical engineering and adds to the problem a new dimension: the distance to the electrode. Therefore, the simulation now has two independent dimensions (time and space) that will define the concentration gradients through solving Fick’s second law of diffusion. In his work [18], Brown used a point-cell method to solve the diffusion-gradient problem for a binary redox system, aimed at the application of Microsoft Excel for the interpretation of cyclic voltammetry. This work will provide MATLAB

^{®}simulations coded like the method that Brown applied in Excel, the point method. The point method consists in the approximation of the concentration gradient to a cell-based grid defined by a time and a distance axis. Spatial and temporal indexes were defined as i and j, respectively, creating a cell grid constituted by those same dimensions. The aim is to conduct a discretization of the differential equations to obtain an expression that will calculate the concentration of a cell based on the prior concentrations in space and time. The discretization of Fick’s second Law is presented with Equations (9) and (10).

^{−1}) as an auxiliary factor (Equation (11)) that groups up the diffusion coefficient of the species, D (cm

^{2}/s); and the time and distance increments, Δt (s) and Δx (cm), respectively obtained by dividing t

_{CV}and L (the thickness of the diffuse layer) by the total number of increments of those dimensions, n

_{t}(Equation (12)) and n

_{x}, (Equation (13)).

_{ox}and J

_{red}(mol cm

^{−2}s

^{−1}) will be rewritten from their classical definition to an expression that includes the available concentrations one distance-increment away from the electrode surface, ${C}_{ox}\left(1\mathit{\Delta}x,t\right)$ and ${C}_{red}\left(1\mathit{\Delta}x,t\right)$ (Equations (19) and (20)).

_{F}(A/g), can be finally obtained with Equation (25), where m is the mass of the electrode’s active material.

_{C}(A/g) to i

_{F}, obtaining the total current density, i

_{T}, in Equation (27).

## 3. Simulation

#### 3.1. Operating Condition

#### 3.2. Simulated Cyclic Voltammetry

_{pC}and i

_{pC}, respectively defining the oxidation and reduction reactions.

_{min}) of −0.7 V from where the current intensity remains null until a steep increase is noted at 0.2 V. At this stage, the active material is behaving as an anode onto which the oxidation is governed by the Butler–Volmer kinetics until it reaches the anodic current peak of 0.32 V, which represents the maximum that the active material can oxidize at the given conditions. Once the Co(OH)

_{2}surface is fully oxidized into CoOOH, no more OH

^{−}ions are being required by the active material to react with. With the oxidative step completed, the current intensity will decrease until the potential sweep is reversed or until a secondary oxidation is activated by a higher potential. Considering the construction of this system and its limitations, no more reactions will occur during the forward sweep and the electrochemical response will be diffusion controlled. Upon reaching the maximum/ending potential (ψ

_{max}) of 0.5 V, the potential sweep is reversed, and the backward step begins. With the potential now decreasing linearly over time under a constant rate of −ν, the Butler–Volmer kinetics govern again and force the active material to be reduced from CoOOH back to Co(OH)

_{2}. The current decreases to the cathodic current peak of 0.2 V, representing the full reduction of the active material. Analogous to the forward sweep, the backward sweep is then controlled by diffusion and increases gradually as the negative current density increases until the potential reaches ψ

_{min}again.

_{2}. It is a simple comprehensive way of observing how Butler–Volmer kinetics can demonstrate the electrochemical behavior of a redox couple such as Co(OH)

_{2}/CoOOH when combined with Fick’s 2nd Law of diffusion. The concentration gradient of the reductant occurs in a relatively analogous process to the oxidant, but with a contrary driving force. Considering the different diffusion coefficients that define the diffusion behavior of the electrolyte species, the gradients will differ slightly in the shape of the colored stain evidently visible. The gradients can be also displayed on the distance planes (Figure 5a,b).

_{max}. In the case of the oxidant (Figure 5a), the minimum concentration is higher as the distance to the electrode increases because the redox processes only occur at the active material’s surface or at its near-surface. The opposite situation occurs for the reductant (Figure 5b).

#### 3.3. Sensitivity Analysis

#### 3.4. Capacitive Contribution

_{2}/CoOOH system, an extra step of the simulation was added, in which the ideal capacitor approximation stated by Equation (26) was assumed, leading to the pure capacitive CV curve observed (Figure 7).

#### 3.5. Simulation vs. Reality

_{2}O

_{2}before being submitted to a CV test in 1 M KOH at 25 °C and 50 mV s

^{−1}.

_{2}/CoOOH redox potentials. A similar approach to the current peaks is also visible at those potentials, with the distance between peaks being more extensive in the simulated curve than the experimental one. This may have to do with the effectively activated surface of the electrode that participates and enables the redox reactions, usually experimentally obtained through functionalization methods such as the oxidative treatment of the metal foam, as described in the previous chapter. Additionally, the overall area of the experimental curve is considerably wider than the simulation’s, most likely as a consequence of other diffusion processes, and possibly other reactions and kinetic phenomena, that are not covered within the simulation models. Unconsidered secondary reactions are also known to occur in these types of pseudocapacitive systems. In fact, the small current peak noticeable at the end of the forward sweep of the experimental curve is possibly related to water electrolysis usually under the form of oxygen evolution reactions (OER). This reaction is essentially the effective oxidation of water under one of four possible mechanisms for alkaline media [29]. Therefore, the addition of the OER was included in the simulation as an extra reaction. According to the Pourbaix diagram of water, for highly alkaline solutions such as of the simulation’s and experiment’s case of 1 M (KOH), the standard potential for the oxidation of water is set at a ψ

_{0}of 0.401 V. The resulting simulation is in Figure 9.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 6.**Sensitivity analysts of CV simulation for (

**a**) scan rate, (

**b**) temperature, and (

**c**) reaction rate constant.

Parameter | Definition | Value | Units | Reference |
---|---|---|---|---|

$z$ | Number of transferred electrons | 1 | – | – |

$\alpha $ | Charge transfer coefficient | 0.5 | – | – |

$R$ | Gas law constant | 8.31451 | J mol^{−1} K^{−1} | – |

$F$ | Faraday’s constant | 96486 | C/mol | – |

$T$ | Temperature | 298.15 | K | [20] |

${\psi}_{0}$ | Standard potential | 0.25 | V | [20] |

${\psi}_{min}$ | Minimum potential | −0.7 | V | [20] |

${\psi}_{max}$ | Maximum potential | 0.5 | V | [20] |

$\nu $ | Scan rate | 0.05 | V/s | [20] |

${t}_{CV}$ | CV duration | 24 | s | Calc. Equation (1) |

$S$ | Electrode area | 0.154 | cm^{2} | [5] |

$m$ | Mass of Co(OH)_{2} nanofoam | 0.01 | g | [20] |

$L$ | Diffuse layer thickness | 0.3019 | cm | Calc. Equation (18) |

$H$ | Helmholtz layer | $1\times {10}^{-7}$ | cm | [21] |

$\mathit{\Delta}t$ | Time increment | 0.08 | s | Calc. Equation (12) |

$\mathit{\Delta}x$ | Distance increment | 0.0062 | cm | Calc. Equation (13) |

${D}_{ox}$ | Diffusion coefficient of OH^{−} | $5.27\times {10}^{-5}$ | cm^{2}/s | [22] |

${D}_{red}$ | Diffusion coefficient of H_{2}O | $2.23\times {10}^{-5}$ | cm^{2}/s | [23] |

${C}_{ox}^{b}$ | Bulk concentration of OH^{−} | 0.001 | mol/cm^{3} | [20] |

${C}_{red}^{b}$ | Bulk concentration of H_{2}O | 0 | mol/cm^{3} | [20] |

${k}_{0}$ | Heterogeneous rate constant | 0.01 | cm/s | [24] |

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**MDPI and ACS Style**

Carvalho, G.G.; Eugénio, S.; Silva, M.T.; Montemor, M.F.
Simulation of the Electrochemical Response of Cobalt Hydroxide Electrodes for Energy Storage. *Batteries* **2022**, *8*, 37.
https://doi.org/10.3390/batteries8040037

**AMA Style**

Carvalho GG, Eugénio S, Silva MT, Montemor MF.
Simulation of the Electrochemical Response of Cobalt Hydroxide Electrodes for Energy Storage. *Batteries*. 2022; 8(4):37.
https://doi.org/10.3390/batteries8040037

**Chicago/Turabian Style**

Carvalho, Gabriel Garcia, Sónia Eugénio, Maria Teresa Silva, and Maria Fátima Montemor.
2022. "Simulation of the Electrochemical Response of Cobalt Hydroxide Electrodes for Energy Storage" *Batteries* 8, no. 4: 37.
https://doi.org/10.3390/batteries8040037