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Journal of Applied Electrochemistry

, Volume 40, Issue 5, pp 967–980 | Cite as

Mathematical modeling of AC electroosmosis in microfluidic and nanofluidic chips using equilibrium and non-equilibrium approaches

  • Jiří Hrdlička
  • Petr Červenka
  • Michal Přibyl
  • Dalimil Šnita
Original Paper

Abstract

AC electroosmotic micropumps are suggested to be powerful tools for electrolyte dosing in various micro- and nanofluidic systems. In this paper, we compare two modeling approaches for studying the AC electroosmosis in the following micro and nanochannel systems: (i) a traveling-wave AC pump with a spatially continuous wave of electric potential applied on a planar boundary, (ii) a traveling-wave AC pump with a wave of electric potential applied on a set of discrete planar electrodes, and (iii) an AC pump with a set of non-planar electrodes. The equilibrium approach is based on the use of capacitor–resistor boundary conditions for electric potential and the slip boundary conditions for velocity at electrode surfaces. The non-equilibrium approach uses the mathematical model based on the Poisson equation and the non-slip boundary conditions. We have observed discrepancies between the predictions given by the both models and then we have identified their possible reasons. The comparison of the equilibrium and non-equilibrium results further showed three important actualities: (a) how the equilibrium model overestimates or underestimates the net velocity, (b) how the velocity maxima in the frequency characteristics can be shifted, if the equilibrium model assumptions are not satisfied, (c) the parametric region where the equilibrium model is applicable. Because the data are obtained in a dimensionless form, they can be exploited for AC electroosmotic studies. We discuss the limitations of the equilibrium and non-equilibrium models and compare selected predictions with available experimental data.

Keywords

Electrokinetics Microfluidics Nanofluidics AC electroosmosis Traveling-wave Mathematical modeling 

List of symbols

A

Amplitude (V)

c

Concentration (mol m−3)

CD

Capacitance of EDL (F m−2)   \(C_{D} = \varepsilon/\lambda_{D}\)

D

Diffusivity (2 × 10−9 m2 s−1)

f

Frequency (s−1)

F

The Faraday constant (96,485 C mol−1)

g

Gap width (m)   g = x m+1 L x m R

h

Electrode height (m)

H

Height of a periodic segment (m)

J

Ion flux intensity (mol m−2s−1)

k

Wave number (m−1)   k = 2π/L

L

Length of a periodic segment (m)

Le

Electrode width (m)   L e  = x m R  − x m L

n

Number of electrodes

nFx

Number of finite elements in the x-direction

nFy

Number of finite elements in the y-direction

n

Normal unit vector

p

Pressure (Pa)

q

Electric charge density (C m−3)

R

Molar gas constant (8.314 J K−1mol−1)

t

Time (s)

t

Tangential unit vector

T

Temperature (298.15 K)

Tt

Period of the electric signal (s)   T t  = f −1

v

Horizontal component of velocity (m s−1)

v

Net velocity (m s−1)

v

Velocity (m s−1)

w

Electric potential wave velocity (m s−1)   w = L/T t  = ω/k

x

Spatial coordinate (m)

y

Spatial coordinate (m)

Greek symbols

α

Phase of an AC signal

\(\varepsilon\)

Electrolyte permitivity (6.9503 × 10−10 F m−1)

φ

Electric potential (V)

η

Dynamic viscosity (0.001 Pa s)

λD

The Debye length (m)   \(\lambda_{D}^{2}=\frac{\varepsilon D}{\sigma}\)

ψ

Complex electric potential (V)

ρ

Density (1,000 kg m−3)

σ

Specific conductivity (S m−1)   \({\sigma}=2 c_{\circ} D \frac{F^{2}}{RT}\)

ω

Angular frequency (s−1)   ω = 2 πf

Dimensionless criteria

Ra

The Rayleigh number   \(\hbox{Ra} = \frac{\varepsilon}{\eta D}\left(\frac{RT}{F}\right)^{2}= 0.2294\)

Sc

The Schmidt number   \(\hbox{Sc} =\frac{\eta}{\rho D} =500\)

\({\tilde{\lambda}}_{D}\)

EDL simplex   \({\tilde{\lambda}}_{D} =\lambda_{D}/L\)

Superscripts

*

Complex conjugate

Dimensionless

Time averaged

+

Cation

Anion

±

Either + or

e

Electrode

L

Left boundary of the electrode

R

Right boundary of the electrode

C

Center of the electrode

Subscripts

o

Characteristic value

m

Index of electrode

slip

At the slip plane

Notes

Acknowledgements

The authors thank for the support by the grant of the GAAV ČR (KAN208240651), by the grant of the MŠMT ČR (MSM 6046137306), by the grant MPO ČR (Pokrok 1H-PK/24), and by the grant GAČR (GD 104/08/H055).

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Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  • Jiří Hrdlička
    • 1
  • Petr Červenka
    • 1
  • Michal Přibyl
    • 1
  • Dalimil Šnita
    • 1
  1. 1.Department of Chemical EngineeringInstitute of Chemical Technology, PraguePrague 6Czech Republic

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