# Travelling-Wave Electrophoresis, Electro-Hydrodynamics, Electro-Rotation, and Symmetry-Breaking of a Polarizable Dimer in Non-Uniform Fields

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Polarization

_{0}(s):

_{0}(s) must vanish both for $s\to 0$ and $s\to \infty $.

_{0}(normalized with respect to the dimensionless volume $\pi /3$ of the dimer [31]):

_{0}(0) defined in Equation (11) by substituting the exact solution given in Equation (9), which renders (see [53], 3.523.3 & 3.557.3):

## 3. Electro-Rotation

_{1}(s) in Equation (14) (symmetric with respect to $\upsilon $), can be found in a similar manner to Equation (6) by enforcing the Robin boundary condition $\partial {\varphi}_{1}/\partial \upsilon =-i\left({\mu}^{2}+{\upsilon}^{2}\right)\Omega {\varphi}_{1}$ on $\upsilon =1$, resulting in the following second-order inhomogeneous ODE for A

_{1}(s):

_{0}) acting in the x and y directions, such that the total field is given by:

## 4. Traveling-Wave Electrophoresis

_{0}is the modified Bessel function of the first kind and zero order. The forcing reference amplitude is denoted by E

_{0}, k represents its wave number, $\phi $ is an arbitrary phase angle, and $\omega $ is the forcing frequency. The particular form of Equation (25) is selected so that under the long-wave approximation $\left(k\to 0,\phi =\pi /2\right)$ one gets $\mathrm{Re}\{{\overline{{\rm X}}}_{TW}\left(z,r,0\right)\}={E}_{0}z$, representing a time-harmonic axisymmetric ambient uniform field.

^{n}can be also expressed in term of a Legendre polynomial ${P}_{n}\left(\tilde{\mathsf{\eta}}\right)$ as ${R}^{n}{P}_{n}\left(\tilde{\mathsf{\eta}}\right)$, where R

^{2}= r

^{2}+ z

^{2}and $\tilde{\mathsf{\eta}}=co{s}^{-1}\left(z/R\right)$. Note that on the z-axis (r = 0), R = z and $\tilde{\mathsf{\eta}}=0$. Thus, the polynomial z

^{n}can be considered as the limiting value of an axisymmetric harmonic function of (z, r) evaluated on r = 0. Hence, following Equations (1), (2), and (26), one gets for $\upsilon >0$ and $\mu \to 0$:

_{DEP,}which is exerted on the dimer by the travelling-wave ambient field given in Equations (25) and (26), can then be expressed following [5,54] in terms of the above multipoles as:

_{m,}are defined in Equation (26). Note that only the odd-order (2m + 1) multipoles of Equation (33) contribute to the DEP force in Equation (34).

## 5. Induced-Charge Electroosmosis

## 6. Electro-Hydrodynamics of a Particle Next to a Wall

## 7. Induced-Charge Electrophoresis of a Janus Dimer

_{d}represents the surface area of the dimer. Substituting the values of the metric coefficients ${h}_{\mu}=1/\left(1+{\mu}^{2}\right),{h}_{\varphi}=\mu /\left(1+{\mu}^{2}\right)$, and $\partial z/\partial \mu =-2\mu /\left(1+{\mu}^{2}\right)$ in Equation (63), yields for ${S}_{d}=2\pi $:

_{0}(s), as given by Equation (10) (see also Figure 2a), namely ${A}_{0}\left(s\right)tanh\left(s\right)/s=s\left(1-2s\right){e}^{-s}sinh\left(s\right)/4$, which leads to:

## 8. Discussion and Summary

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

A_{n} | Complex function forming the integrand solution of the general time-dependent potential, see Equation (2) |

a | Radius of the sphere |

C | Constant (Section 7) |

D | Diffusivity of the symmetric monovalent electrolyte |

${\overrightarrow{d}}_{eff}$ | Effective dipole |

E_{0} | Forcing amplitude of the excitation |

E(t) | Forcing electric field |

E^{4} | Biharmonic operator |

$\hat{\mathrm{e}}$ | Unit vector |

h | Metric coefficient |

I_{0} | Modified Bessel function of the first kind |

J_{0} | Bessel function of the first kind |

k | Wave number |

n | Normal unit vector |

P_{n} | Legendre polynomial |

R | Spherical radius |

r | Polar radius in the x–y plane, see Figure 3 |

S_{d} | Surface area of the dimer |

s | Integration variable, e.g., Equation (2) |

t | Time |

U | Velocity component |

$\overrightarrow{v}$ | Velocity vector |

x | Axial coordinate, see Figure 1 |

y | Lateral coordinate |

z | The dimer axisymmetric coordinate, see Figure 1 |

$\epsilon $ | Electric permittivity |

$\zeta $ | Euler–Riemann zeta function |

$\tilde{\mathsf{\eta}}$ | Spherical angle |

$\eta $ | Dynamic viscosity |

$\Theta \xb0$ | Angular velocity |

${\lambda}_{0}$ | Nano-metric EDL thickness |

$\left(\begin{array}{c}\mu \\ \upsilon \\ \phi \end{array}\right)$ | Orthogonal tangent-sphere coordinate system |

$\xi $ | Induced potential |

$\overrightarrow{\tau}$ | Electrostatic torque |

$\varphi $ | Total electric potential |

${X}_{TW}$ | Ambient axisymmetric electric forcing of a non-homogenous travelling-wave (TW) excitation (Section 4) |

$\mathsf{\chi}$ | Harmonic function, a component of the total electric potential in the far- field |

$\mathsf{\psi}$ | Stream function |

$\mathsf{\Omega}$ | RC dimensionless frequency |

$\omega $ | Frequency |

AC | Alternating current |

EDL | Electric double layer |

DC | Direct current |

DEP | Dielectrophoresis |

EHD | Electro-hydrodynamic |

HS | Helmholtz–Smoluchowski |

ICEO | Induced-charge electroosmosis |

ICEP | Induced-charge electrophoresis |

PNP | Poisson–Nernst–Planck |

RC | Resistance–capacitance circuit |

ROT | Electro-rotation |

TW | Travelling wave |

TWDEP | Travelling-wave dielectrophoresis |

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**Figure 1.**Schematic description of the problem of a dimer that is composed of two geometrically identical spheres in free space as in Section 2, Section 3, Section 4 and Section 5 and Section 7. The dimer is subjected to: a uniform AC electric field acting in the z direction (Section 2, Section 3, Section 4 and Section 5 and Section 7), a uniform AC electric field acting in the x direction (Section 3), and a non-homogeneous axisymmetric travelling wave propagating along the z direction (Section 4). The two spheres are identically conductive in Section 2, Section 3, Section 4 and Section 5, and in Section 7 the lower sphere is coated by a thin dielectric layer.

**Figure 2.**The solution of Equation (8) of Section 2 for (

**a**) $\Omega =0$, (

**b**) $\Omega =0.5$, (

**c**) $\Omega =1$, and (

**d**) $\Omega =10$, and where the exact solution of $\Omega =0$ is given by Equation (9). The asymptotic solution is of Equation (10).

**Figure 3.**Schematic description of the problem in Section 6 of a spherical particle next to a wall (z = 0), which is subjected to a uniform DC electric field acting in the z direction.

**Figure 4.**The (

**a**) contours of the Stokes stream function and (

**b**) velocity vectors around the spherical particle placed next to a wall at z = 0 of Section 6, and which is subjected to a uniform DC electric field acting in the z direction. The velocity-vector field modulus was adjusted for better viewing.

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**MDPI and ACS Style**

Miloh, T.; Avital, E.J.
Travelling-Wave Electrophoresis, Electro-Hydrodynamics, Electro-Rotation, and Symmetry-Breaking of a Polarizable Dimer in Non-Uniform Fields. *Micromachines* **2022**, *13*, 1173.
https://doi.org/10.3390/mi13081173

**AMA Style**

Miloh T, Avital EJ.
Travelling-Wave Electrophoresis, Electro-Hydrodynamics, Electro-Rotation, and Symmetry-Breaking of a Polarizable Dimer in Non-Uniform Fields. *Micromachines*. 2022; 13(8):1173.
https://doi.org/10.3390/mi13081173

**Chicago/Turabian Style**

Miloh, Touvia, and Eldad J. Avital.
2022. "Travelling-Wave Electrophoresis, Electro-Hydrodynamics, Electro-Rotation, and Symmetry-Breaking of a Polarizable Dimer in Non-Uniform Fields" *Micromachines* 13, no. 8: 1173.
https://doi.org/10.3390/mi13081173