# Non-Linear Electrohydrodynamics in Microfluidic Devices

## Abstract

**:**

## 1. Introduction

_{e}the volumetric density of the free charge, W the volumetric density of the electroquasistatic energy, and α

_{1}, α

_{2}, …, α

_{m}the material properties. The first term is the Coulomb force density originating from free charges. The second is the dielectrophoresis force density originating from the bounded (paired) charges. Techniques that exploit the first forcing term are referred to as electrophoresis; techniques that exploit mainly the second are referred to as dielectrophoresis.

## 2. Theory

**E**= −∇φ, ɛ the electric permittivity of the medium, and ρ

_{e}the same as that defined in Equation (1), the volumetric density of the free charge.

_{e}—both are functions of space—hence the electric field.

_{k}is the volumetric concentration, z

_{k}the valence, and F the Faraday constant.

_{k}accounts for the contributions from electromigration, diffusion and convection with the fluid flow. μ

_{k}is the electro-mobility and D

_{k}the diffusivity, D

_{k}= μ

_{k}RT according to Nernst-Einstein relation.

## 3. Interfacial Electrohydrodynamics

_{i}, ɛ

_{d}and ɛ

_{a}are electric permittivity of the insulating layer, the droplet and the surrounding liquid, respectively. This expression shows the force magnitude is proportional to ɛ

_{i}V

^{2}/d. The value within the (.) in Equation (6) approaches one, the maximum, when ɛ

_{a}/ ɛ

_{d}≪ 1 and d / D ≪ 1. In practice, the droplets are either conductive or highly polarizable [5] (e.g., aqueous based), and the surrounding medium is usually a non-polar solvent, that is, ɛ

_{d}≫ ɛ

_{i}and ɛ

_{a}≈ ɛ

_{i}; the thickness of the insulating layer is orders of magnitude smaller than the size of the droplets, D ≫ d. If we write δ ≡ (ɛ

_{a}/ ɛ

_{i})(d / D), Equation (6) can be re-written as

## 4. Non-Linear Electrokinetics

_{∥}the electric field component parallel to the surface. For the charged species, a common practice is to assume a constant μ

_{ep}, the electrophoretic mobility, and express the particle migration velocity as

**U**= μ

_{ep}

**E**. Since the electric impact on both the solid-liquid surface (electroosmosis) and the charged species (electrophoresis) is linear with respect to the electric field, this type of applications and modeling treatment is called linear electrokinetics.

## 5. Conclusions: Enabling the Systems

## Acknowledgments

## References

- Muller, RS. MEMS: Quo vadis in century XXI. Microelectron. Eng
**2000**, 53(1–4), 47–54. [Google Scholar] - Reyes, DR; Iossifidis, D; Auroux, PA; Manz, A. Micro total analysis systems. 1. Introduction, theory, and technology. Anal.Chem
**2002**, 74, 2623–2636. [Google Scholar] - Levy, U; Shamai, R. Tunable optofluidic devices. Microfluid. Nanofluid
**2008**, 4, 97–105. [Google Scholar] - Zeng, J; Korsmeyer, FT. Principles of droplet electrohydrodynamics for lab-on-a-chip. Lab Chip
**2004**, 4, 265–277. [Google Scholar] - Fair, RB. Digital microfluidics: Is a true lab-on-a-chip possible? Microfluid. Nanofluid
**2007**, 3, 245–281. [Google Scholar] - Pollack, MG; Fair, RB; Shenderov, AD. Electrowetting-based actuation of liquid droplets for microfluidic applications. Appl. Phys. Lett
**2000**, 77(11), 1725–1726. [Google Scholar] - Peykov, V; Quinn, A; Ralston, J. Electrowetting: A model for contact-angle saturation. Colloid Polym. Sci
**2000**, 278, 789–793. [Google Scholar] - Verheijen, HJJ; Prins, MWJ. Reversible electrowetting and trapping of charge: Model and experiments. Langmuir
**1999**, 15, 6616–6620. [Google Scholar] - Mugele, F; Baret, J. Electrowetting: From basics to applications. J. Phys. Condens. Matter
**2005**, 17, R705–R774. [Google Scholar] - Quilliet, C; Berge, B. Electrowetting: A recent outbreak. Curr. Opin. Colloid Interface Sci
**2001**, 6, 34–39. [Google Scholar] - Probstein, RF. Physicochemical Hydrodynamics; Wiley: New York, NY, USA, 1994. [Google Scholar]
- Koo, J; Kleinstreuer, C. Liquid flow in microchannels: Experimental observations and computational analyses of microfluidics effects. J. Micromech. Microeng
**2003**, 13, 568–579. [Google Scholar] - Hu, G; Li, D. Multiscale phenomena in microfluidics and nanofluidics. Chem. Eng. Sci
**2007**, 62, 3443–3454. [Google Scholar] - Haus, HA; Melcher, JR. Electromagnetic Fields and Energy; Prentice-Hall: Englewood Cliffs, NJ, USA, 1989. [Google Scholar]
- Leal, LG. Laminar Flow and Convective Transport Processes: Scaling Principles and Asymptotic Analysis; Butterworth-Heinemann: Oxford, UK, 1992. [Google Scholar]
- Collins, RT; Harris, MT; Basaran, OA. Breakup of electrified jets. J. Fluid Mech
**2007**, 588, 75–129. [Google Scholar] - Sista, R; Hua, Z; Thwar, P; Sudarsan, A; Srinivasan, V; Eckhardt, A; Pollack, M; Pamula, V. Development of a digital microfluidic platform for point of care testing. Lab Chip
**2008**, 8, 2091–2104. [Google Scholar] - Zeng, J. Modeling and simulation of electrified droplets and its application to computer-aided design of digital microfluidics. IEEE Trans. Comput. Aid. Des. Integr. Circ. Syst
**2006**, 25(2), 224–233. [Google Scholar] - Walker, SW; Bonito, A; Nochetto, RH. Mixed finite element method for electrowetting on dielectric with contact line pinning. Interface. Free Bound
**2010**, 12, 85–119. [Google Scholar] - Eck, C; Fontelos, M; Grün, G; Klingbeil, F; Vantzos, O. On a phase-field model for electrowetting. Interface. Free Bound
**2009**, 11, 259–290. [Google Scholar] - Gascoyne, PRC; Vykoukal, JV. Dielectrophoresis-based sample handling in general-purpose programmable diagnostic instruments. Proc. IEEE
**2004**, 92(1), 22–42. [Google Scholar] - Jones, TB; Gunji, M; Washizu, M. Dielectrophoretic liquid actuation and nanodroplet formation. J. Appl. Phys
**2001**, 89(3), 1441–1448. [Google Scholar] - Sretavan, D; Chang, W; Keller, C; Kliot, M. Microscale surgery on single axons. Neurosurgery
**2005**, 57(4), 635–646. [Google Scholar] - Pohl, HA; Crane, JS. Dielectrophoresis of cells. Biophys. J
**1971**, 11, 711–727. [Google Scholar] - Melcher, JR; Taylor, GI. Electrohydrodynamics: A review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech
**1969**, 1, 111–146. [Google Scholar] - Saville, DA. Electrohydrodynamics: The taylor-melcher leaky-dielectric model. Annu. Rev. Fluid Mech
**1997**, 29, 27–64. [Google Scholar] - Schultz, GA; Corso, TN; Prosser, SJ; Zhang, S. A fully integrated monolithic microchip electrospray device for mass spectrometry. Anal. Chem
**2000**, 72(17), 4058–4063. [Google Scholar] - Killeen, K; Yin, H; Udiavar, S; Brennen, R; Juanitas, M; Poon, E; Sobek, D; van de Goor, T. Chip-MS: A polymeric microfluidic device with integrated mass-spectrometer interface. Micro Total Anal. Syst
**2001**, 331–332. [Google Scholar] - Dukhin, SS. Electrokinetic phenomena of the second kind and their applications. Adv. Colloid Interface Sci
**1991**, 35, 173–196. [Google Scholar] - Wang, Y-C; Stevens, AL; Han, J. Million-fold preconcentration of proteins and peptides by nanofluidic filter. Anal. Chem
**2005**, 77(14), 4293–4299. [Google Scholar] - Kim, SJ; Wang, Y-C; Han, J. Nonlinear electrokinetic flow pattern near nanofluidic channel. Micro Total Anal. Syst
**2006**, 1, 522–524. [Google Scholar] - Comiskey, B; Albert, JD; Yoshizawa, H; Jacobson, J. An electrophoretic ink for all-printed reflective electronic displays. Nature
**1998**, 394(6690), 253–255. [Google Scholar] - Beunis, F; Strubbe, F; Neyts, K; Bert, T; De Smet, H; Verschueren, A; Schlangen, L. P-39: Electric field compensation in electrophoretic ink display. In Proceedings of the Twenty-fifth International Display Research Conference—Eurodisplay 2005; Edinburgh, UK, 19–22 2005; pp. 344–345. [Google Scholar]
- Strubbe, F; Verschueren, ARM; Schlangen, LJM; Beunis, F; Neyts, K. Generation current of charged micelles in nonaqueous liquids: Measurements and simulations. J. Colloid Interface Sci
**2006**, 300, 396–403. [Google Scholar] - Hsu, MF; Dufresne, ER; Weitz, DA. Charge stabilization in nonpolar solvents. Langmuir
**2005**, 21, 4881–4887. [Google Scholar] - Hayes, RA; Feenstra, BJ. Video-speed electronic paper based on electrowetting. Nature
**2003**, 425, 383–385. [Google Scholar] - Chakrabarty, K; Su, F. Digital Microfluidic Biochips: Synthesis, Testing, and Reconfiguration Techniques; CRC Press: Boca Raton, FL, USA, 2006. [Google Scholar]
- Chakrabarty, K; Fair, RB; Zeng, J. Design tools for digital microfluidic biochips: Towards functional diversification and more than Moore. IEEE Trans.CAD Integr. Circ. Syst
**2010**, 29(7), 1001–1017. [Google Scholar]

**Figure 1.**Example of droplet-based digital microfluidics architecture. Above is an elevation view showing the layered structure of the chip. Below is a diagram illustrating the system (Adapted from [4]).

**Figure 2.**Simulation of droplet separation by EWOD. The top two figures illustrate the device configuration. Electric voltages are applied to all four electrodes embedded in the insulating material. The bottom left figure shows transient simulation solution. It illustrates the process of separating one droplet into two via EWOD. The bottom right figure shows the electric potential distribution inside the device. The color indicates the electric potential; the iso-potential surfaces are also drawn. The image shows the electric field is absent within the droplet body indicating the droplet is either conductive or highly polarizable.

**Figure 3.**Simulation of dielectrophoresis driven axon migration. The set of small images on the left shows a transient simulation of single axon migration under an electric field generated by a pin electrode. The image on the right is a snapshot of a simulation where two axons are fused by dielectrophoresis using a pin electrode. Axons are outlined in white. Also shown are the iso-potential curves.

**Figure 4.**Transient sequence of the Taylor cone formation: simulation and experiment comparison. Experimental images are shown in the top row. Simulation results are shown in the bottom row. Their correspondence is indicated by the vertical alignment (Adapted from [4]).

**Figure 5.**Equilibrium vs. non-equilibrium electroosmosis. Left: concentration polarization of the equilibrium electroosmotic flow. The normalized curves are counter-ion concentration, co-ion concentration, and the velocity of the electroosmotic flow, obtained from numerical simulation. Right: schematic representation of the concentration polarization of the non-equilibrium electroosmosis. The curves are the normalized concentrations of the counter-ions and co-ions.

**Figure 6.**Simulation of charge screening effect using a parallel-plate cell. Top-left image shows the electric current as function of time and driving voltage, top-right image shows the evolution of the species concentration as function of time and space, the bottom image shows the electric current readout after switching the applied voltage.

**Figure 7.**Transient simulation of electrohydrodynamic instability and the development of the cellular convective flow pattern.

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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Zeng, J.
Non-Linear Electrohydrodynamics in Microfluidic Devices. *Int. J. Mol. Sci.* **2011**, *12*, 1633-1649.
https://doi.org/10.3390/ijms12031633

**AMA Style**

Zeng J.
Non-Linear Electrohydrodynamics in Microfluidic Devices. *International Journal of Molecular Sciences*. 2011; 12(3):1633-1649.
https://doi.org/10.3390/ijms12031633

**Chicago/Turabian Style**

Zeng, Jun.
2011. "Non-Linear Electrohydrodynamics in Microfluidic Devices" *International Journal of Molecular Sciences* 12, no. 3: 1633-1649.
https://doi.org/10.3390/ijms12031633