Journal of Applied Electrochemistry

, Volume 40, Issue 5, pp 967–980

# 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).

## References

1. 1.
Ramos A, Morgan H, Green NG et al (1998) J Phys D 31:2338
2. 2.
Ajdari A (2000) Phys Rev E 61:R45
3. 3.
Campisi M, Accoto D, Dario P (2005) J Chem Phys 123:204724
4. 4.
Garcia-Sanchez P, Ramos A, Green G et al (2006) IEEE Trans Dielectr Electr Insul 13:670
5. 5.
Green NG, Ramos A, Gonzalez A et al (2002) Phys Rev E 66:026305
6. 6.
Mpholo M, Smith CG, Brown ABD (2003) Sens Actuators B 92:262
7. 7.
Studer V, Pepin A, Chen Y et al (2004) Analyst 129:944
8. 8.
Probstein RF (1994) Physicochemical hydrodynamics: an introduction. Wiley, New York
9. 9.
Green NG, Ramos A, Morgan H (2000) J Phys D 33:632
10. 10.
Bazant MZ, Ben YX (2006) Lab Chip 6:1455
11. 11.
Burch D, Bazant MZ (2008) Phys Rev E 77:055303(R)
12. 12.
Urbanski JP, Thorsen T, Levitan JA et al (2006) Appl Phys Lett 89:143508
13. 13.
Urbanski JP, Levitan JA, Burch DN et al (2007) J Colloid Interface Sci 309:332
14. 14.
Cahill BP, Heyderman LJ, Gobrecht J et al (2004) Phys Rev E 70:036305
15. 15.
Cahill BP, Heyderman LJ, Gobrecht J et al (2005) Sens Actuators B 110:157
16. 16.
Ramos A, Gonzalez A, Garcia-Sanchez P et al (2007) J Colloid Interface Sci 309:323
17. 17.
Ejsing L, Smistrup K, Pedersen CM et al (2006) Phys Rev E 73:037302
18. 18.
Mortensen NA, Olesen LH, Belmon L et al (2005) Phys Rev E 71:056306
19. 19.
Ramos A, Morgan H, Green NG et al (2005) J Appl Phys 97:084906
20. 20.
Squires TM, Bazant MZ (2004) J Fluid Mech 509:217
21. 21.
Olesen LH, Bruus H, Ajdari A (2006) Phys Rev E 73:056313
22. 22.
Kim BJ, Yoon SY, Sung H J et al (2007) J Appl Phys 102:074513
23. 23.
Loucaides N, Ramos A, Georghiou GE (2007) Microfluid Nanofluid 3:709
24. 24.
Khan T, Reppert PM (2005) J Colloid Interface Sci 290:574
25. 25.
Wang XM, Wu JK (2006) J Colloid Interface Sci 293:483
26. 26.
Pribyl M, Snita D, Marek M (2008) Multiphysical modeling of DC and AC electroosmosis in micro- and nanosystems. In: Petrone G, Cammarata G (eds) Recent advances in modelling and simulation. I-Tech Education and Publishing, ViennaGoogle Scholar
27. 27.
Cervenka P, Pribyl M, Snita D (2009) Microelectron Eng 86:1333
28. 28.
Kilic MS, Bazant MZ, Ajdari A (2007) Phys Rev E 75:021502
29. 29.
Kilic MS, Bazant MZ, Ajdari A (2007) Phys Rev E 75:021503
30. 30.
Storey BD, Edwards L R, Kilic MS et al (2008) Phys Rev E 77:036317
31. 31.
Levitan JA, Devasenathipathy S, Studer V et al (2005) Colloids Surf A 267:122
32. 32.
González A, Ramos A, Green NG et al (2000) Phys Rev E 61:4019
33. 33.
Postler T, Slouka Z, Svoboda M et al (2008) J Colloid Interface Sci 320:321
34. 34.
Deen WM (1998) Analysis of transport phenomena. Oxford University Press, New YorkGoogle Scholar

## Authors and Affiliations

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