# Numerical analysis of field-modulated electroosmotic flows in microchannels with arbitrary numbers and configurations of discrete electrodes

## Authors

- First Online:

DOI: 10.1007/s10544-010-9450-1

- Cite this article as:
- Chao, K., Chen, B. & Wu, J. Biomed Microdevices (2010) 12: 959. doi:10.1007/s10544-010-9450-1

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## Abstract

The formation of an electric double layer and electroosmosis are important theoretic foundations associated with microfluidic systems. Field-modulated electroosmotic flows in microchannels can be obtained by applying modulating electric fields in a direction perpendicular to a channel wall. This paper presents a systematic numerical analysis of modulated electroosmotic flows in a microchannel with discrete electrodes on the basis of the Poisson equation of electric fields in a liquid–solid coupled domain, the Navier–Stokes equation of liquid flow, and the Nernst-Planck equation of ion transport. These equations are nonlinearly coupled and are simultaneously solved numerically for the electroosmotic flow velocity, electric potential, and ion concentrations in the microchannel. A number of numerical examples of modulated electroosmotic flows in microchannels with discrete electrodes are presented, including single electrodes, symmetric/asymmetric double electrodes, and triple electrodes. Numerical results indicate that chaotic circulation flows, micro-vortices, and effective fluid mixing can be realized in microchannels by applying modulating electric fields with various electrode configurations. The interaction of a modulating field with an applied field along the channel is also discussed.

### Keywords

Electric double layerModulated electroosmotic flowMicrochannelMicro-vorticesFluid mixing## 1 Introduction

Microfluidic systems are studied in a number of fields associated with fluid mechanics, biology, chemistry, electricity, and heat transfer. The formation of an electric double layer (EDL) and electroosmosis are important theoretic foundations associated with microfluidic systems. Electroosmotic flows (EOFs) have become an increasingly efficient fluid transport mechanism in microfluidic systems. The magnitude, direction, and flow profile of an EOF in a microchannel can be effectively controlled by applying an electric field perpendicular to a solid wall. Controlling electroosmotic flows may enhance the solution separation by capillary electrophoresis. It is clearly of interest to be able to directly and dynamically control the polarity and the wall zeta potential at the aqueous/channel interface (Lee et al. 1990). The wall potential can be modulated anytime during the analysis to control electroosmotic flows within the microfluidic network (Buch et al. 2001) referred to as the “flow field effect transistor” (flow FET) (Schasfoort 1999; Ajdari 1995; Ajdari 1996; Chen and Conlisk 2008; Daiguji et al. 2008). Chaotic electroosmotic flows effectively enhance fluid mixing in microchannels (Erickson and Li 2002). Both spatial and temporal control of the wall potential can be achieved by applying transverse electric fields (Hayes and Ewing 1992). Such electric fields can be applied with the aid of electrodes embedded beneath the solid–liquid interface and electrically insulated from the liquid. These electrodes can be either continuous or discrete for different application purposes. Most of the researches on field-modulated EOFs have focused on theoretical models (Petsev 2005), experimental studies (Sniadecki et al. 2004), and fabrication techniques (Hayes et al. 1993; Polson and Hayes 2000). Generally speaking, analytic solutions may not be available for a complex microfluidic system, especially in the cases of discrete electrodes with different sizes, layouts, and voltages. A numerical analysis of electroosmotic flows in microchannel based on a slip model and a two-capacitor model was presented (Van Theemsche et al. 2003). The two-capacitor model is based on the assumption of uniformly continuous wall potential of a microchannel. These approximate models are not applicable for the cases of discrete electrodes which create non-uniform or discontinuous wall potential. Another concern is that the wall potential (or wall charge density) is specified as a boundary condition in the numerical modeling of EOF without the consideration of an insulated layer between the electrodes and the aqueous electrolyte (Daiguji et al. 2004; Mirbozorgi et al. 2006; Mirbozorgi et al. 2007; Lynn et al. 2008; Chen and Cho 2008). However, how the wall potential was created was not reported. The insulated layer is necessary in some cases to prevent an electrochemical reaction on the wall. The induced wall potential is not known in advance when the voltage is applied on an electrode surface. The insulated layer must be taken into account for modulated electroosmotic flows. Therefore, a sophisticated numerical model is necessary. An electroosmotic flow system involves the multi-physics phenomena of liquid flow, electricity, and electrolyte ion transportation. The Navier–Stokes equation, Poisson equation, and Nernst-Planck equation are coupled and are simultaneously solved in the numerical analysis of electroosmotic flows. The objective of this study is to present a systematic numerical analysis to predict the electroosmotic flow behavior and performance of an electroosmotic flow system with arbitrary numbers and configurations of discrete electrodes. A number of numerical examples will be presented to demonstrate the functions of a field-modulated electroosmotic flow and the efficiency of liquid mixing in microchannel.

## 2 Governing equations and boundary conditions

### 2.1 Governing equations

*is the fluid flow velocity vector,*

**V***p*is the liquid pressure,

*ρ*is the liquid density,

*μ*is the liquid viscosity,

*ρ*

_{e}is the net charge density, and

*ρ*

_{e}

*is the electric force acting on the fluid. The electric field*

**E***is related to the potential as follows:*

**E***ψ*is the induced potential of a field-modulated EDL and

*φ*is the potential of the applied electric field along the channel. The EDL potential

*ψ*is governed by the Poisson equation as follows:

*ε*

_{r},

*ε*

_{0}are the dielectric constants in the mediums and in vacuum,

*e*is the elementary charge,

*z*

_{i}is the valence of the ion, and

*n*

_{i}is the ionic number concentration of the electrolyte solution. The applied potential

*φ*is governed by the Laplace equation as follows:

*λ*is the electric conductivity of the bulk liquid, assumed to be a constant. The ionic concentration

*n*

_{i}is governed by the Nernst-Planck equation as follows:

**J**

_{i}is the ionic flux of the ion species

*i*,

*D*

_{i}is the diffusivity of the ions,

*T*is the absolute temperature, and

*k*

_{b}is the Boltzmann constant. Liquid mixing is governed by the transport equation of the chemical species, written as follows:

*C*,

*D*

_{c}are the species concentration and diffusivity, respectively. The governing equations of the fluid motion, electric field, and ion transport are coupled. The scopes of this study are stated as following.

- (1)
The native wall zeta potential has been extensively studied, it will not be considered in current study.

- (2)
Faradaic current on channel wall will not be considered.

- (3)
The wall potential induced by the modulating electric field is below the steric limit (less than 200

*mV*) (Kilic et al. 2007a; Kilic et al. 2007b); therefore, the steric effects will not be taken into account in the present study.

### 2.2 Boundary conditions

*ABCD*) are shown in Fig. 1(b).

*V*

_{elec}is the potential of the electrode surface at location

*x*and

*φ*(

*x*) is the potential of the applied electric field along the channel at the electrode location

*x*.

*V*

_{e}is the difference between the electrode potential and the potential of the channel center, which is referred to as the modulating electric potential responsible for the locally induced wall potential and the ion concentration variation.

The channel inlet and outlet are sufficiently far away from the electrodes, where *n* is the out- normal vector of the boundary.

*A*′

*B*′

*C*′

*D*′ given by Eq. 5, the boundary conditions are specified as follows:

*A*′

*B*′

*C*′

*D*′ are as follows:

*A*′

*B*′

*C*′

*D*′ are as follows:

It is equivalent to zero normal ionic flux (*J*_{i})_{n} = 0 (Mirbozorgi et al. 2006), where *n*_{0} is the ion number concentration of the bulk fluid.

## 3 Numerical results and discussion

*KCL* aqueous solutions at room temperature *T* = 298 *K* are used as the working fluid. The dielectric constant of the channel wall and the solutions are *ε*_{1} = 2.0 and *ε*_{2} = 78.5, respectively; the ion diffusivity is \( {D_{+} } = {D_{-} } = 2 \times {10^{ - 9}}\;{m^2}/s \). The liquid density is *ρ* = 1000 *kg*/*m*^{3}, and the viscosity is *μ* = 10^{−3}
*N s*/*m*^{2}. Generally, the widths of the electrode and the channel are considerably smaller than the channel length. The typical data of electrokinetic flows in the microchannel are as follows: channel width *h* = 10 *μm*, the electrode width *a* = *h*, the thickness of the dielectric layer *δ* = *h*/10, and the channel length *L* = 20 *h*. The electric potential applied in the channel inlet is *φ*_{1} = 20 *V*; therefore, the applied electric field along the channel is *E*_{0} = 10^{5}
*V*/*m*. The multi-physics module of the computational software COMSOL is used for carrying out the numerical analysis.

### 3.1 Single electrode model

*κh*= 32, where

*κh*is the ratio of the channel width to the EDL thickness, where \( \kappa = \frac{1}{{{\lambda_D}}} = \sqrt {{\frac{{2{n_0}{e^2}{z^2}}}{{\varepsilon {k_b}T}}}} \) and

*λ*

_{D}is the characteristic thickness of EDL. The numerical solutions of two types of grids agree well, and the non-structural grids are used in the following study. It can be seen that the induced wall potential increases with an increase in the modulating potential |

*V*

_{e}| and shows asymmetry. The

*K*

^{+}ions are attracted to the wall by a negative wall charge, and

*Cl*

^{−}ions are expelled away from the wall; the local wall ion concentrations are shown in Fig. 3(b), (c). The effect of the thickness of the dielectric layer of the wall on the induced potential and the ion concentration are shown in Fig. 4. It can be seen that the induced potential and the ion concentration increase with a decrease of

*κδ*(the ratio of the thickness of the dielectric layer to the characteristic thickness of the EDL). It is also found that the induced potential and the ion concentration are independent of the microchannel width while modulating potential is fixed.

### 3.2 Multi-electrode model

*Q*

_{0}=

*U*

_{0}

*h*and

*U*

_{0}= −

*ε*

_{2}

*ε*

_{0}

*E*

_{0}

*ζ*

_{0}/

*μ*. It can be seen that the flow rate increases with an increase in the modulating electric field

*V*

_{e}. With fixed modulating potential

*V*

_{e}, the flow rate of one electrode is the same as that of triple electrodes, but smaller than that of the two electrodes. The flow rates of the two electrodes are independent of the electrode layout in the present cases. In order to assess the liquid mixing efficiency, a mixing coefficient is introduced. The section-averaged concentration is calculated as \( C_{a} {\left( x \right)} = \frac{1} {h}{\int_h {C{\left( {x,y} \right)}dy} } \), and then the mean square deviation of the concentration is calculated as \( \sigma {\left( x \right)} = {\sqrt {\frac{1} {h}{\int_h {{\left[ {C{\left( {x,y} \right)} - C_{a} {\left( x \right)}} \right]}^{2} dy} }} } \) and the mixing coefficient is defined as \( \alpha = 1 - {{{{\sigma_{out}}}} \left/ {{{\sigma_{in}}}} \right.} \). The liquid mixing efficiencies in the microchannel are shown in Fig. 11. It can be seen that the liquid mixing performance of the triple-electrode model is better than that of the single-electrode and double-electrode models. Vortices in the triple-electrode model stirs the flow streamline considerably more effectively than those in the double-electrode model. It is also found that the increasing modulating voltage increases the flow rate of the microchannels but decreases the liquid mixing efficiency.

## 4 Conclusion

This paper presents the systematic numerical analysis of a field-modulated electroosmotic flow in a microchannel with discrete electrodes based on the nonlinearly coupled Poisson equation, the Navier–Stokes equation, and the Nernst-Planck equation. These equations are numerically solved by using a multi-physics module of the computational software COMSOL. A number of numerical examples of the modulated electroosmotic flows in microchannels with discrete electrodes are presented. It is found that chaotic circulation flows, micro-vortices, and effective fluid mixing can be realized in microchannels by the interaction of the applied electric field along the channel with the modulating electric field with an optimal design of discrete electrodes.