# Scattering of Metal Colloids by a Circular Post under Electric Fields

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## Abstract

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## 1. Introduction

## 2. Analysis of the Physical Problem

#### 2.1. Particle Repulsion with Its Image Dipole

#### 2.2. ICEO Repulsion from Insulating Walls

#### 2.3. Dielectrophoresis and Dipolophoresis

#### 2.4. Particle Trajectory and Comparisons between Mechanisms

## 3. Numerical Simulations of the Trajectories

- (A)
- Electric field parallel to the fluid flow. We imposed boundary conditions of zero normal current density ($\partial \varphi /\partial n=0$) at the cylinder surface and at upper and lower planes (see the geometry in Figure 1). Dirichlet boundary conditions were applied at the entrance and exit so that the applied electric field was equal to ${E}_{0}$.
- (B)
- Electric field perpendicular to the fluid flow. We imposed boundary conditions of zero normal current density ($\partial \varphi /\partial n=0$) at the cylinder surface and at the entrance and exit. Dirichlet boundary conditions were applied at upper and lower planes.

#### 3.1. Electric Field Parallel to the Fluid Flow

#### 3.2. Electric Field Perpendicular to the Fluid Flow

#### 3.3. Particle Deviations

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Dipole-Dipole Repulsion from a Cylinder

**Figure A1.**3D domain for computing the reflections by the cylinder of both the electric fields of the dipoles and the velocity field of the stresslets.

**Figure A2.**Nondimensional force on dipole versus distance to the cylinder surface. Comparison between numerical force and the theoretical approximation according to Equation (1).

## Appendix B. ICEO Interaction with a Cylinder

**Figure A3.**Particle drift velocity due to ICEO flow reflected on a cylinder versus the distance to the cylinder wall.

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**Figure 1.**Schematics of the problem. Particles enter the channel from the left side driven by the fluid flow with velocity ${\mathit{u}}_{\mathbf{0}}$. There is an ac electric field applied in the system. The particle trajectories are affected by the presence of an insulating post. The particle-post interaction arises from different mechanisms of electrical origin.

**Figure 2.**Particle trajectories for an applied electric field parallel to flow direction: (

**a**) Low frequency and $\mathsf{\Lambda}=1$. (

**b**) Low frequency and $\mathsf{\Lambda}=0.5$. (

**c**) High frequency. The colors represent the magnitude of the particle velocity in units of ${v}_{0}$.

**Figure 3.**Particle trajectories for an applied electric field perpendicular to flow direction: (

**a**) low frequency and $\mathsf{\Lambda}=1$; (

**b**) low frequency and $\mathsf{\Lambda}=0.5$; (

**c**) high frequency. The colors represent the magnitude of the particle velocity in units of ${v}_{0}$.

**Figure 4.**Zoom of particle trajectories for an applied electric field perpendicular to flow direction in the case of low frequency and $\mathsf{\Lambda}=1$. The colors represent the magnitude of the particle velocity in units of ${v}_{0}$.

**Figure 5.**Deviation ${y}_{f}-{y}_{i}$ versus ${y}_{i}$ at $N=50$ for two cases: (

**A**) low frequency and $\mathsf{\Lambda}=1$; (

**B**) low frequency and $\mathsf{\Lambda}=0.5$.

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**MDPI and ACS Style**

Flores-Mena, J.E.; García-Sánchez, P.; Ramos, A.
Scattering of Metal Colloids by a Circular Post under Electric Fields. *Micromachines* **2023**, *14*, 23.
https://doi.org/10.3390/mi14010023

**AMA Style**

Flores-Mena JE, García-Sánchez P, Ramos A.
Scattering of Metal Colloids by a Circular Post under Electric Fields. *Micromachines*. 2023; 14(1):23.
https://doi.org/10.3390/mi14010023

**Chicago/Turabian Style**

Flores-Mena, José Eladio, Pablo García-Sánchez, and Antonio Ramos.
2023. "Scattering of Metal Colloids by a Circular Post under Electric Fields" *Micromachines* 14, no. 1: 23.
https://doi.org/10.3390/mi14010023