# Stochastic Density Functional Theory on Lane Formation in Electric-Field-Driven Ionic Mixtures: Flow-Kernel-Based Formulation

## Abstract

**:**

## 1. Introduction

## 2. Basic Formalism

#### 2.1. Primitive Model

#### 2.2. Stochastic Dft: Compact Matrix Forms

## 3. Our Aim

## 4. Correlation Functions Determined by the Stochastic Dft

#### 4.1. Stationary Condition of Correlation Functions

#### 4.2. Obtained Forms of Stationary Correlation Functions

## 5. Lane Formation in Terms of Charge–Charge Correlation Function

#### 5.1. Asymptotic Behavior of Charge–Charge Correlations

#### 5.2. Charge–Charge Correlations on 2D Cross Section of the 3D Primitive Model

## 6. Summary and Conclusions

## Funding

## Conflicts of Interest

## Appendix A. Deterministic Dft: Introduction of Flow Kernels

## Appendix B. Linear Stability Analysis Based on the Deterministic Dft

#### Appendix B.1. Dispersion Relation

**Figure A1.**Schematics of the electrophoresis-induced shear. It is found from the schematic on the right that the shear rate $\dot{\gamma}$ induced by cations (or cation-driven-shear rate) is evaluated as $\dot{\gamma}\sim \mathcal{D}zE/R$ when oppositely charged colloids with their effective diameter of R pass each other.

**Figure A2.**Schematics of advection velocities with fluctuating flows in the y-direction. We can consider four cases of the fluctuating velocities generated under (

**a**) the cation-driven-shear and (

**b**) the anion-driven-shear.

#### Appendix B.2. Derivation of the Relation $\stackrel{~}{\epsilon}$ ~ zEσ Using an Expression of the Flow Kernel G(r) for Sheared Colloids

## Appendix C. Details on the Derivation of Stationary Correlation Functions

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**Figure 1.**A schematic of the 2D primitive model of binary ionic mixture with a static electric field $\mathit{E}$ applied in the x-direction. The z-valent cations and anions are modeled by equisized charged hard spheres of diameter $\sigma $ immersed in a dielectric medium with dielectric constant $\u03f5$ at a temperature T.

**Figure 2.**A schematic of lane formation in a binary ionic mixture. The green and orange lanes represent aligned segregation bands of cations and anions, respectively. Correspondingly, the positive and negative signs seen on the lanes indicate that each lane is a mesoscopically charged object. The wavelengths, ${\lambda}_{x}^{\ast}$ and ${\lambda}_{y}^{\ast}$, in x-and y-directions are related to wavenumbers as ${\lambda}_{x}^{\ast}=2\pi /{k}_{x}^{\ast}$ and ${\lambda}_{y}^{\ast}=2\pi /{k}_{y}^{\ast}$ (i.e., Equation (A17)), respectively. In this study, these wavenumbers are determined by Equations (34) and (37) when considering point charges.

**Figure 3.**A schematic of the 3D primitive model in Cartesian coordinates illustrates a binary ionic mixture confined between two parallel plates. While the y-axis is perpendicular to these plates, the electric field is applied in the x-axis.

**Figure 4.**The real-space representation ${\mathcal{C}}_{qq}^{\mathrm{st}}(x,y)$ of the charge-charge correlation function at $zE\sigma =1.0$ is shown using heat maps where the length scale is in units of diameter $\sigma $. We obtain the real-space correlation function from performing the 2D inverse Fourier transform of ${\mathcal{C}}_{qq}^{\mathrm{st}}\left(\mathit{k}\right)/\left(2\overline{n}\right)$ given by Equations (24) and (25). The remaining parameter set of $(\tilde{\mathcal{E}},\phantom{\rule{0.166667em}{0ex}}\overline{\kappa}\sigma ,\phantom{\rule{0.166667em}{0ex}}\varphi )$ is (

**a**) $(0.484,\phantom{\rule{0.166667em}{0ex}}1.55,\phantom{\rule{0.166667em}{0ex}}0.05)$, (

**b**) $(0.484,\phantom{\rule{0.166667em}{0ex}}1.62,\phantom{\rule{0.166667em}{0ex}}0.055)$, (

**c**) $(0.0,\phantom{\rule{0.166667em}{0ex}}1.55,\phantom{\rule{0.166667em}{0ex}}0.05)$, and (

**d**) $(0.0,\phantom{\rule{0.166667em}{0ex}}1.62,\phantom{\rule{0.166667em}{0ex}}0.055)$.

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Frusawa, H. Stochastic Density Functional Theory on Lane Formation in Electric-Field-Driven Ionic Mixtures: Flow-Kernel-Based Formulation. *Entropy* **2022**, *24*, 500.
https://doi.org/10.3390/e24040500

**AMA Style**

Frusawa H. Stochastic Density Functional Theory on Lane Formation in Electric-Field-Driven Ionic Mixtures: Flow-Kernel-Based Formulation. *Entropy*. 2022; 24(4):500.
https://doi.org/10.3390/e24040500

**Chicago/Turabian Style**

Frusawa, Hiroshi. 2022. "Stochastic Density Functional Theory on Lane Formation in Electric-Field-Driven Ionic Mixtures: Flow-Kernel-Based Formulation" *Entropy* 24, no. 4: 500.
https://doi.org/10.3390/e24040500