# Electric Double Layers with Surface Charge Regulation Using Density Functional Theory

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## Abstract

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## 1. Introduction

## 2. Charge Regulation

## 3. The Grand Thermodynamic Potential

## 4. Higher Order Electrostatic Correlations

## 5. Molecular Interactions, Solvent Polarity Effects and Dielectric Permittivity

## 6. Results and Discussion

#### 6.1. Solvent Effects

#### 6.2. Effect of Screening Correlations on EDL Structure within the Primitive Model

#### 6.3. Effect on Fluid Flow

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

EDL | Electrostatic double layer |

cDFT | Classical Density functional theory |

HS | Hard sphere |

LJ | Lennard-Jones |

MSA | Mean spherical approximation |

HNC | Hypernetted chain |

SC | Screening correlations |

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**Figure 1.**Fluid charge density ${\rho}_{e}$ for three different cases of solvent-wall interactions: the blue curve is for ${\epsilon}_{s-w}=0$ (solvophobic), the black is for ${\epsilon}_{s-w}=1{k}_{B}T$ (neutral, where the energy is the same as the bulk LJ attraction). The red curve is for ${\epsilon}_{s-w}=2{k}_{B}T$ (solvophilic).

**Figure 2.**Electrostatic potential for three different cases of solvent-wall interactions: the blue curve is for ${\epsilon}_{s-w}=0$ (solvophobic), the black is for ${\epsilon}_{s-w}=1{k}_{B}T$ (neutral, where the energy is the same as the bulk LJ attraction). The red curve is for ${\epsilon}_{s-w}=2{k}_{B}T$ (solvophilic).The results shown correspond to the same charges as for Figure 1.

**Figure 3.**Classical density functional theory (cDFT) (

**solid lines**) and Poisson-Boltzmann (

**dashed lines**) calculations of electric double layer (EDL) structure for a primitive model electrolyte with $+2$ valence cations and $-1$ anions, both with $0.3$ nm diameter. The surface charge is $-0.1$ C/m${}^{2}$. (

**a**) Electrostatic potential versus x, the distance from the wall, for low (${10}^{-2}$ M, blue lines) and high (1 M, black lines) bulk concentration. (

**b**) Cation (

**black lines**) and anion (

**red lines**) concentration profiles for the 1 M case.

**Figure 4.**Geometry of a nanofluidic slit channel. Two wells with electrolytes (

**blue**) are connected by a channel fabricated out of fused silica that is several millimeters in the x direction, several micrometers in the z direction and $\sim 100$ nm in the y direction. The surface charge is on the top (

**pink**) and bottom (

**red**) of the device.

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**MDPI and ACS Style**

Gillespie, D.; Petsev, D.N.; van Swol, F.
Electric Double Layers with Surface Charge Regulation Using Density Functional Theory. *Entropy* **2020**, *22*, 132.
https://doi.org/10.3390/e22020132

**AMA Style**

Gillespie D, Petsev DN, van Swol F.
Electric Double Layers with Surface Charge Regulation Using Density Functional Theory. *Entropy*. 2020; 22(2):132.
https://doi.org/10.3390/e22020132

**Chicago/Turabian Style**

Gillespie, Dirk, Dimiter N. Petsev, and Frank van Swol.
2020. "Electric Double Layers with Surface Charge Regulation Using Density Functional Theory" *Entropy* 22, no. 2: 132.
https://doi.org/10.3390/e22020132