# Interplay between Forced Convection and Electroconvection during the Overlimiting Ion Transport through Anion-Exchange Membranes: A Fourier Transform Analysis of Membrane Voltage Drops

^{*}

## Abstract

**:**

## 1. Introduction

- Region I (quasi-ohmic): the current density increases linearly with the voltage until ${i}_{lim}$ is reached. In this region, ion flux is proportional to the applied electric field.
- Region II (plateau): in this region, an increase in the applied voltage implies a subtle or almost absent increase in current density. The scarcity of ions near the diluted membrane surface is the main cause for larger membrane resistances.
- Region III (overlimiting): if the applied voltage is further increased, transport phenomena other than diffusion and migration arise, and current densities above ${i}_{lim}$ can be registered.

## 2. Materials and Methods

#### 2.1. Membranes and Electrodialysis Cell

#### 2.2. Electrochemical Characterization of the Membrane-Electrolyte Systems

#### 2.3. Treatment of Chronopotentiometric Data and Fourier Transform Analysis

## 3. Results and Discussion

#### 3.1. Effect of Forced Convection, Applied Current Density and Membrane Type on the Current–Voltage Characteristics: ${i}_{lim}$ and ${l}_{plateau}$

#### 3.2. Amplitude of Membrane Voltage Oscillations

#### 3.3. FT Analysis of Membrane Voltage Signals

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ED | Electrodialysis |

FT | Fourier Transform |

FFT | Fast Fourier Transform |

## Appendix A

**Figure A2.**Chronopotentiometric response obtained with AMV-N membrane and NaCl as electrolyte at (

**a**) 0 rpm, (

**b**) 200 rpm, (

**c**) 400 rpm, (

**d**) 600 rpm.

**Figure A3.**Chronopotentiometric response obtained with HC-A membrane and NaCl as electrolyte at (

**a**) 0 rpm, (

**b**) 200 rpm, (

**c**) 400 rpm, (

**d**) 600 rpm.

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**Figure 2.**Treatment and standardization of chronopotentiograms for the FFT analysis. (

**a**) Original chronopotentiogram before treatment, (

**b**) after treatment, and (

**c**) resulting spectrum obtained after the signal treatment via Fast Fourier Transform. Dashed lines represent the average trend of the data.

**Figure 3.**Current–voltage curves obtained at various stirring rates. (

**a**) Homogeneous membrane, AMV-N; (

**b**) Heterogeneous membrane, HC-A.

**Figure 4.**Comparison between the characteristic parameters at various stirring rates for both membranes: (

**a**) Limiting current density, and (

**b**) plateau length.

**Figure 5.**Current–voltage curves obtained at 0 and 200 rpm and pH evolution of the solution in the diluted ED compartment vs. ${U}_{m}$. (

**a**) AMV-N, (

**b**) HC-A.

**Figure 6.**Evolution of the amplitude of membrane voltage oscillations at various levels of stirring rate. (

**a**) AMV-N with NaCl as electrolyte, (

**b**) HC-A with NaCl as electrolyte. Evolution of the amplitude of membrane voltage oscillations with NaCl and Na${}_{2}$SO${}_{4}$ at 400 rpm. (

**c**) AMV-N, (

**d**) HC-A.

**Figure 7.**Fourier Transform of the membrane AMV-N at (

**a**) 0 rpm, (

**b**) 200 rpm, (

**c**) 400 rpm, (

**d**) 600 rpm.

**Figure 8.**Fourier Transform of the membrane HC-A at (

**a**) 0 rpm, (

**b**) 200 rpm, (

**c**) 400 rpm, (

**d**) 600 rpm.

**Figure 10.**(

**a**) Chronopotentiometric response obtained with AMV-N membrane at 1.5 $\times \phantom{\rule{3.33333pt}{0ex}}{i}_{lim}$ at 400 rpm, (

**b**) Fourier Transform, (

**c**) Chronopotentiometric response obtained with HC-A membrane at 1.5 $\times \phantom{\rule{3.33333pt}{0ex}}{i}_{lim}$ at 400 rpm, (

**d**) Fourier Transform.

**Table 1.**Experimental conditions applied in the electrochemical characterization of the membrane-electrolyte systems.

Membrane | Electrolyte | Stirring Rate | Reynolds Number |
---|---|---|---|

AMV-N | NaCl | 0 rpm | 0 |

AMV-N | NaCl | 200 rpm | 1498 |

AMV-N | NaCl | 400 rpm | 2995 |

AMV-N | NaCl | 600 rpm | 4493 |

HC-A | NaCl | 0 rpm | 0 |

HC-A | NaCl | 200 rpm | 1498 |

HC-A | NaCl | 400 rpm | 2995 |

HC-A | NaCl | 600 rpm | 4493 |

**Table 2.**Experimental conditions applied in the evaluation of the effect of the type of counter-ion on the development of electroconvection.

Membrane | Electrolyte | Stirring Rate | Reynolds Number |
---|---|---|---|

AMV-N | NaCl | 400 rpm | 2995 |

AMV-N | Na${}_{2}$SO${}_{4}$ | 400 rpm | 2971 |

HC-A | NaCl | 400 rpm | 2995 |

HC-A | Na${}_{2}$SO${}_{4}$ | 400 rpm | 2971 |

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**MDPI and ACS Style**

Hernández-Pérez, L.; Martí-Calatayud, M.C.; Montañés, M.T.; Pérez-Herranz, V.
Interplay between Forced Convection and Electroconvection during the Overlimiting Ion Transport through Anion-Exchange Membranes: A Fourier Transform Analysis of Membrane Voltage Drops. *Membranes* **2023**, *13*, 363.
https://doi.org/10.3390/membranes13030363

**AMA Style**

Hernández-Pérez L, Martí-Calatayud MC, Montañés MT, Pérez-Herranz V.
Interplay between Forced Convection and Electroconvection during the Overlimiting Ion Transport through Anion-Exchange Membranes: A Fourier Transform Analysis of Membrane Voltage Drops. *Membranes*. 2023; 13(3):363.
https://doi.org/10.3390/membranes13030363

**Chicago/Turabian Style**

Hernández-Pérez, Lorena, Manuel César Martí-Calatayud, Maria Teresa Montañés, and Valentín Pérez-Herranz.
2023. "Interplay between Forced Convection and Electroconvection during the Overlimiting Ion Transport through Anion-Exchange Membranes: A Fourier Transform Analysis of Membrane Voltage Drops" *Membranes* 13, no. 3: 363.
https://doi.org/10.3390/membranes13030363