# Asymmetrical Induced Charge Electroosmotic Flow on a Herringbone Floating Electrode and Its Application in a Micromixer

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Methods

#### 2.1. Micromixer Design

^{3}to 1.52 × 10

^{5}are tested. The mixing efficiency is simultaneously calculated under different element numbers. As shown in Figure 1f, the mixing efficiency varies slightly with the increasing element numbers beyond 1.10 × 10

^{5}. Therefore, a grid system with 1.10 × 10

^{5}hexahedral elements is selected as the suitable grid system in terms of the accuracy and efficiency of simulation. We can also see from Figure 1f that the mixing efficiency for the herringbone floating electrode is better than the Ridge and Vee type floating electrode when the inlet flow velocity and the bulk conductivity are set to be 500 μm/s and 0.001 S/m under the conditions of A = 5 V, f = 400 Hz. It can be ascribed to the higher transverse slip velocity generated above the herringbone floating electrode under certain conditions. Therefore, we choose the herringbone floating electrode microstructure for further investigation.

#### 2.2. Mathematical Model

_{f}denotes the bulk conductivity, and

**n**represents the normal vector in the interface of the bulk and floating electrode, pointing from the electrode into the bulk electrolyte.

_{f}and w are the bulk permittivity and the angular frequency of the applied electric field, δ = C

_{D}/C

_{S}signifies the surface physical capacitance ratio of the diffuse layer C

_{D}= ε

_{f}/λ

_{D}to the stern layer C

_{S}, and ϕ

_{b}is the transient potential values at the metal surface. Here we introduced the complex phasor amplitude of each electrical field variable as denoted by a tilde for analytical convenience [56], e.g., $\varphi \left(t\right)=A\mathrm{cos}\left(\omega t+\theta \right)=\mathrm{Re}\left(A{e}^{j\theta}{e}^{j\omega t}\right)=\mathrm{Re}\left(\tilde{\varphi}{e}^{j\omega t}\right)$, where, Re( ) is the real part of ( ). At low frequency, there is no displacement current running through the double-layer capacitor skin. For the impenetrability of the ionic species, the sum of diffusion current and ohmic current in the bulk is equal to another one in the diffusion double layer.

**E**

_{t}denote the dynamic viscosity of aqueous media and the tangential field component on the floating electrode surface. In addition, the time-average non-linear electroosmotic slip velocity in AC oscillation can be expressed by:

**u**are hydraulic pressure and fluid velocity vector, respectively.

#### 2.3. Numerical Simulation

_{0}is prescribed to the both inlets for forward transport incoming fluid flow. Meanwhile, the pressure for the outlet flow is zero. Based on the above calculated electric field distribution, the slip velocity u

_{slip}can be obtained and employed on the herringbone floating electrode sequence. The microchannel sidewalls surface expect the electrodes to be imposed as non-slip.

_{f}= 10 μS/cm, the dynamic viscosity is 0.001 Pa·s, and the diffusivity of the fluid sample is 2 × 10

^{−9}m

^{2}/s.

#### 2.4. Evaluation of the Mixing Efficiency

## 3. Results and Discussion

#### 3.1. The Microstream Driven by Induced Charge Electroosmotics (ICEO) in the Channel

#### 3.2. The Electroosmotic Flow Velocity near the Floating Electrode

#### 3.3. Mixing Performance of the Device with Different Number of Herringbone Floating Electrode Pairs

#### 3.4. Parametric Effect on Mixing Performance

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic of induced charge electroosmotic (ICEO) flow on the polarizable metal strip and the ICEO slip velocity on the floating electrode with some deviation from the middle of driving electrodes. (

**a**) The formation of the electric double layer on the floating electrode under the imposed external electrical field; (

**b**) the asymmetrical ICEO microvortices above the polarizable floating electrode surface; (

**c**–

**e**) the Ridge/Vee/Herringbone type floating electrode microstructure; (

**f**) the mesh independency test and the mixing efficiency at the outlet for the three types of metal strip with single electrode pair under the conditions of A = 5 V and f = 400 Hz.

**Figure 2.**Geometry of proposed micromixer based ICEO. (

**a**) 3D schematic diagram of the device with herringbone floating electrode sequence; (

**b**,

**c**) the fluidic sample motion on the left and right herringbone floating electrode surface under the asymmetrical ICEO vortex; (

**d**) the specific dimensions of the microfluidic mixer device.

**Figure 3.**The fluid microstream at different cross section. (

**a**) the definition of cross section 1–6; (

**b**–

**d**) the fluid motions on the left herringbone floating electrode at the cross section 1–3; (

**e**–

**g**) the fluid flow on the right herringbone floating electrode at the cross section 4–6.

**Figure 4.**The frequency dependency of maximum slip velocity on the herringbone floating electrode at different voltage input and liquid conductivity. (

**a**–

**d**) the maximum slip velocity vs frequency when the voltage intensities are 5 V, 10 V, 15 V and 20 V at the liquid conductivity σ = 0.001 S/m, 0.005 S/m, 0.010 S/m.

**Figure 5.**The mixing performance of the micromixer with a pair of herringbone floating electrodes. (

**a**) The effect of voltage intensity on mixing performance at f = 300 Hz, u = 500 μm/s; (

**b**) the frequency dependency of mixing performance at A = 5 V, u = 500 μm/s; (

**c**) the relationship between the mixing performance and inlet flow at A = 5 V, f = 300 Hz; (

**d**) the mixing performance under different liquid conductivity at A = 5 V, f = 300 Hz and u = 500 μm/s.

**Figure 6.**Mixing performance of the device with different number of herringbone floating electrode pairs. (

**a**–

**h**) The simulation results of the micromixer with different amount of Herringbone floating electrode pair; (

**a**) a pair; (

**b**) two pairs; (

**c**) three pairs; (

**d**) four pairs; (

**e**) five pairs; (

**f**) six pairs; (

**g**) seven pairs; (

**h**) eight pairs; (

**i**) the curve illustrating the mixing performance and the number of herringbone floating electrode pair.

**Figure 7.**Parametric effect on mixing performance. (

**a**–

**g**) the concentration distribution at different cross section of channel at A = 5 V, f = 300 Hz, u = 500 μm/s. (

**b**) The cross section at a range of 500 μm; (

**c**) 1000 μm; (

**d**) 1500 μm; (

**e**) 2000 μm; (

**f**) 2500 μm; (

**g**) 3000 μm; (

**h**) the voltage intensity effect on mixing performance at f = 300 Hz, u = 500 μm/s. (

**i**) the frequency dependency of mixing performance at A = 5V, u = 500 μm/s; (

**j**) the influence of inlet flow on mixing performance at A = 5 V, f = 300 Hz. (

**k**) The relationship of mixing performance and liquid conductivity at A = 5 V, f = 300 Hz, u = 500 μm/s.

Parameter | W1 | W2 | W3 | W4 | W5 | W6 | θ |
---|---|---|---|---|---|---|---|

Value (μm) | 50 | 40 | 150 | 30 | 200 | 100 | 90^{o} |

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## Share and Cite

**MDPI and ACS Style**

Hu, Q.; Guo, J.; Cao, Z.; Jiang, H.
Asymmetrical Induced Charge Electroosmotic Flow on a Herringbone Floating Electrode and Its Application in a Micromixer. *Micromachines* **2018**, *9*, 391.
https://doi.org/10.3390/mi9080391

**AMA Style**

Hu Q, Guo J, Cao Z, Jiang H.
Asymmetrical Induced Charge Electroosmotic Flow on a Herringbone Floating Electrode and Its Application in a Micromixer. *Micromachines*. 2018; 9(8):391.
https://doi.org/10.3390/mi9080391

**Chicago/Turabian Style**

Hu, Qingming, Jianhua Guo, Zhongliang Cao, and Hongyuan Jiang.
2018. "Asymmetrical Induced Charge Electroosmotic Flow on a Herringbone Floating Electrode and Its Application in a Micromixer" *Micromachines* 9, no. 8: 391.
https://doi.org/10.3390/mi9080391