# Impedance-Frequency Response of Closed Electrolytic Cells

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

^{3}), and diffusion coefficient D.

#### 2.1. Equation System

- (i).
- Poisson equation:$${\epsilon}_{e}\frac{{\partial}^{2}\mathsf{\Psi}(x,t)}{\partial {x}^{2}}=-ze{N}_{A}\left[{c}_{+}(x,t)-{c}_{-}(x,t)\right]$$
- (ii).
- Nernst–Planck equations:$${J}_{\pm}(x,t)=-D\left\{\frac{\partial {c}_{\pm}(x,t)}{\partial x}\pm {c}_{\pm}(x,t)\frac{\partial}{\partial x}\left[\frac{ze\mathsf{\Psi}(x,t)}{kT}\right]\right\}$$
^{2}s), $k$ is the Boltzmann constant, and $T$ the absolute temperature. - (iii).
- Continuity equations:$$\frac{\partial {J}_{\pm}(x,t)}{\partial x}=-\frac{\partial {c}_{\pm}(x,t)}{\partial t}$$

#### 2.2. Boundary Conditions

#### 2.3. Dimensionless Variables

## 3. Equation System Solution

## 4. Results

#### 4.1. Steady State

#### 4.1.1. Ionic Concentrations at the Central Plane of Closed Electrolytic Cells

#### 4.1.2. Electric Double Layer Thickness

#### 4.2. Frequency Response

_{B}:

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Equation (27)

#### Appendix A.2. Proof of Equation (58)

#### Appendix A.3. Proof of Equation (61)

## References

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**Figure 1.**Ionic concentration at the central plane of closed electrolytic cells as function of the surface potential for the indicated $L/{\lambda}_{D}$ values.

**Figure 2.**Numerical results for the electric potential profiles in a closed electrolytic cell calculated for $L/{\lambda}_{D}=100$ and the indicated surface potential values.

**Figure 3.**Dependence of the electric double layer thickness values on the surface potential, calculated for the indicated $L/{\lambda}_{D}$ values. Vertical dot lines show surface potential values corresponding to the double layer thickness minima, Equation (58).

**Figure 4.**Spectra of the real part of the impedance (

**a**), its imaginary part (

**b**), and its imaginary part multiplied by the angular frequency (

**c**), calculated for the indicated $L/{\lambda}_{D}$ values and the dimensionless surface potential ${\overline{\mathsf{\Psi}}}_{S}^{0}=8$. Dot and dash-dot straight lines (closed and open cells, respectively) correspond to the following: (

**a**) Real part of the impedance at frequencies between both dispersions, Equation (61) for open or (62) for closed cells; (

**b**) characteristic frequency of the high-frequency dispersion, Equation (60) for closed cells; (

**c**) Low-frequency limit of the imaginary part of the impedance multiplied by the frequency, Equations (63), (64), and (25) for closed or Equation (28) for open cells.

**Figure 5.**Spectra of the real part of the impedance (

**a**), the imaginary part (

**b**), and the imaginary part multiplied by the angular frequency (

**c**), calculated for the indicated dimensionless surface potential values and for $L/{\lambda}_{D}=100$. Dot and dash-dot straight lines (closed and open cells respectively) correspond to the following: (

**a**) Real part of the impedance at frequencies between both dispersions, Equation (61) for open or (62) for closed cells; (

**b**) characteristic frequency of the high-frequency dispersion, Equation (60) for closed cells; (

**c**) low-frequency limit of the imaginary part of the impedance multiplied by the frequency, Equations (63), (64), and (25) for closed or Equation (28) for open cells.

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**MDPI and ACS Style**

López-García, J.J.; Horno, J.; Grosse, C.
Impedance-Frequency Response of Closed Electrolytic Cells. *Micromachines* **2023**, *14*, 368.
https://doi.org/10.3390/mi14020368

**AMA Style**

López-García JJ, Horno J, Grosse C.
Impedance-Frequency Response of Closed Electrolytic Cells. *Micromachines*. 2023; 14(2):368.
https://doi.org/10.3390/mi14020368

**Chicago/Turabian Style**

López-García, José Juan, José Horno, and Constantino Grosse.
2023. "Impedance-Frequency Response of Closed Electrolytic Cells" *Micromachines* 14, no. 2: 368.
https://doi.org/10.3390/mi14020368