Biomedical Microdevices

, Volume 12, Issue 6, pp 959–966

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

Authors

  • Kan Chao
    • Mechanics Department of HuazhongUniversity of Science & Technology, Wuhan National Laboratory for Optoelectronics
    • Mechanics Department of HuazhongUniversity of Science & Technology, Wuhan National Laboratory for Optoelectronics
  • Jiankang Wu
    • Mechanics Department of HuazhongUniversity of Science & Technology, Wuhan National Laboratory for Optoelectronics
Article

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

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

For the purpose of easy fabrication, slit-type microchannels are commonly used in microfluidic chips; the thickness of the channels is considerably less than the width. The flows in the slit-type microchannels can be approximated as two-dimensional flows. A group of discrete electrodes is mounted on a channel wall, electrically insulated from the liquid, as shown in Fig. 1(a). When the microchannel is filled with the aqueous electrolyte and subjected to a longitudinal electric field along the channel and a modulating transverse electric field, a complex electroosmotic flow will be generated.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig1_HTML.gif
Fig. 1

Sketch of field-modulated electroosmotic flows in microchannel with discrete electrodes. (a) The sketch of the channel (b) Boundary conditions

2.1 Governing equations

The continuity equation and the Navier–Stokes equation for incompressible viscous fluids are as follows:
$$ \nabla \cdot {\mathbf{V}} = 0 $$
(1)
$$ \rho \left( {\frac{{\partial {\mathbf{V}}}}{{\partial t}} + \left( {{\mathbf{V}} \cdot \nabla } \right){\mathbf{V}}} \right) = - \nabla p + \mu {\nabla^2}{\mathbf{V}} + {\rho_e}{\mathbf{E}} $$
(2)
where V is the fluid flow velocity vector, p is the liquid pressure, ρ is the liquid density, μ is the liquid viscosity, ρe is the net charge density, and ρeE is the electric force acting on the fluid. The electric field E is related to the potential as follows:
$$ {\mathbf{E}} = - \nabla \left( {\varphi + \psi } \right) $$
(3)
where 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:
$$ \nabla \cdot \left( {{\varepsilon_r}\nabla \psi } \right) = - \frac{{{\rho_e}}}{{{\varepsilon_0}}};\left\{ {\begin{array}{*{20}{c}} {{\rho_e} = 0} & {{\hbox{in}}\;{\hbox{dielectric}}\;{\hbox{layer}}\left( {r = {1}} \right)} \\{{\rho_e} = \sum\limits_i {e\left( {{z_i}{n_i}} \right)} } & {{\hbox{in}}\;{\hbox{liquid}}\;{\hbox{domain}}\left( {r = {2}} \right)} \\\end{array} } \right.\; $$
(4)
where εr, ε0 are the dielectric constants in the mediums and in vacuum, e is the elementary charge, zi is the valence of the ion, and ni is the ionic number concentration of the electrolyte solution. The applied potential φ is governed by the Laplace equation as follows:
$$ \begin{array}{*{20}{c}} {\nabla \cdot \left( {\lambda \nabla \varphi } \right) = 0} & {{\hbox{in}}\,{\hbox{liquid}}\,{\hbox{domain}}} \\\end{array} $$
(5)
where λ is the electric conductivity of the bulk liquid, assumed to be a constant. The ionic concentration ni is governed by the Nernst-Planck equation as follows:
$$ \frac{{\partial {n_i}}}{{\partial t}} + \nabla \cdot {{\mathbf{J}}_i} = 0 $$
(6)
$$ {{\mathbf{J}}_i} = {\mathbf{V}}{n_i} - {D_i}\nabla {n_i} - \frac{{{z_i}e{D_i}}}{{{k_b}T}}{n_i}\nabla \left( {\varphi + \psi } \right) $$
(7)
where Ji is the ionic flux of the ion species i, Di is the diffusivity of the ions, T is the absolute temperature, and kb is the Boltzmann constant. Liquid mixing is governed by the transport equation of the chemical species, written as follows:
$$ \frac{{\partial C}}{{\partial t}} + {\mathbf{V}} \cdot \nabla C = {D_c}{\nabla^2}C $$
(8)
where C, Dc 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. (1)

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

     
  2. (2)

    Faradaic current on channel wall will not be considered.

     
  3. (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

The boundary conditions for the Poisson Eq. 4 in the liquid–solid coupling domain (ABCD) are shown in Fig. 1(b).
$$ \begin{array}{*{20}{c}} {\psi = {V_e} = {V_{elec}} - \varphi (x)} & {{\hbox{on}}\;{\hbox{all}}\;{\hbox{elctrode}}\;{\hbox{surfaces}}\;e - e} \\\end{array} $$
(9)
where Velec 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. Ve 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.
$$ \begin{array}{*{20}{c}} {\frac{{\partial \psi }}{{\partial n}} = 0} & {{\hbox{for}}\;{\hbox{all}}\;{\hbox{non}} - {\hbox{elctrode}}\;{\hbox{surfaces}}\;s - s} \\\end{array} $$
(10)
$$ \begin{array}{*{20}{c}} {\frac{{\partial \psi }}{{\partial n}} = 0,} & {{\hbox{in}}\;{\hbox{channel}}\;{\hbox{inlet}}\;A\,B\;{\hbox{and}}\;{\hbox{outlet}}\;C\,D} \\\end{array} $$
(11)

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

For the applied electric field along the channel in the liquid domain ABCD′ given by Eq. 5, the boundary conditions are specified as follows:
$$ \varphi = {\varphi_1}\;{\hbox{in channel inlet}}\;A\prime B\prime, \;\varphi = 0 \;{\hbox{in channel outlet}}\;C\prime D\prime $$
(12)
$$ \begin{array}{*{20}{c}} {\frac{{\partial \varphi }}{{\partial n}} = 0,} & {\hbox{on insulated channel wall}} \\\end{array} \;B\prime C\prime \;{\hbox{and}}\;A\prime D\prime $$
(13)
The boundary conditions for the Navier–Stokes Eq. 2 in the liquid domain ABCD′ are as follows:
$$ p = 0\;{\hbox{in the channel inlet and outlet}},\,\,V = 0\;{\hbox{on the channel wall}} $$
(14)
The boundary conditions for the Nernst-Planck Eqs. 6 and 7 in the liquid domain ABCD′ are as follows:
$$ {n_i} = {n_0}\;{\hbox{in channel inlet}}\;A\prime\;B\prime\;{\hbox{and}}\;\frac{{\partial {n_i}}}{{\partial n}} = 0\;{\hbox{in}}\;{\hbox{outlet}}\;C\prime\;D\prime $$
(15)
The channel inlet and outlet are far away from the electrodes.
$$ n_{i} = n_{0} \exp {\left( { - \frac{{z_{i} e\psi }} {{k_{b} T}}} \right)}on\,channel\,wall\,B\prime C\prime \,and\,A\prime D\prime $$
(16)

It is equivalent to zero normal ionic flux (Ji)n = 0 (Mirbozorgi et al. 2006), where n0 is the ion number concentration of the bulk fluid.

For the transport equation of the chemical species (18), two liquids A and B enter the channel.
$$ {C_{O{A^\prime}}} = 1.0,\;\;{C_{O{B^\prime}}} = 0.0\;{\hbox{in channel inlet}}\;A\prime\;B\prime $$
(17)
$$ \frac{{\partial C}} {{\partial n}} = 0\,on\,channel\,wall\,B\prime \,C\prime \,and\,A\prime \,D\prime $$
(18)
$$ \frac{{\partial C}} {{\partial n}} = 0\,on\,channel\,outlet\,C\prime \,D\prime $$
(19)

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 ε1 = 2.0 and ε2 = 78.5, respectively; the ion diffusivity is \( {D_{+} } = {D_{-} } = 2 \times {10^{ - 9}}\;{m^2}/s \). The liquid density is ρ = 1000 kg/m3, and the viscosity is μ = 10−3N s/m2. 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 φ1 = 20 V; therefore, the applied electric field along the channel is E0 = 105V/m. The multi-physics module of the computational software COMSOL is used for carrying out the numerical analysis.

3.1 Single electrode model

Structural grids and non-structural grids are used in the numerical analysis to ensure that the numerical solutions are independent of grids, as shown in Fig. 2. The induced wall potential are shown in Fig. 3(a) with different modulating fields, where \( {\zeta_0} = \frac{{{k_b}T}}{{ze}} \approx 25\,mV \), 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 λD 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 |Ve| 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.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig2_HTML.gif
Fig. 2

Numerical grids in the vicinity of the electrode. (a) Structural grids (b) Non-structural grids

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig3_HTML.gif
Fig. 3

Local induced wall potential, ionic concentrations K+ and Cl for various modulating potentials. The electrokinetic diameter κh = 32 and the dielectric layer κδ = 3.2. (a) Local induced wall potential and a comparison of the structural grids and the non-structural grids (b) Local wall ionic concentrations (c) The ionic concentrations profiles in channel section below the electrode

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig4_HTML.gif
Fig. 4

Local induced wall potential, ionic concentrations K+ and Cl for various thickness of the dielectric layer κδ. The modulating potential Ve= −100 ζ0 and the electrokinetic diameter κh = 32. (a) Local induced wall potential (b) Local wall ionic concentrations (c) The ionic concentrations profiles in channel section below the electrode

The asymmetry of the induced wall potential and ion concentrations is due to the interaction of the applied field along the channel and the transverse modulating field, as shown in Fig. 5. It can be seen that the electric potential is symmetrical and concentrated in a thin layer close to the channel wall in cases without the applied electric field along the channel. When an electric field is applied along the channel, the ions move chaotically, resulting in a redistribution of the ion concentration, charge density, and the electric potential, which are asymmetrically extended to the entire channel.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig5_HTML.gif
Fig. 5

Local induced potential contours in the microchannel. The modulating potential Ve= −500 ζ0, the electrokinetic diameter κh = 32 and the dielectric layer κδ = 3.2. (a) Without applied electric fields along the channel (b) With applied electric fields along the channel

Electroosmotic velocity vectors, streamlines, and liquid mixing in the microchannels are shown in Fig. 6. It can be seen that the chaotic circulation flow and a vortex are generated by the local disturbance of the modulating field. The local chaotic flow effectively enhances the liquid mixing in the microchannels.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig6_HTML.gif
Fig. 6

Electroosmotic flow vectors, streamlines, enhanced liquid mixing in the microchannel modulated by electric fields of a single electrode. The electrokinetic diameter κh = 32 and the dielectric layer κδ = 3.2. (a) Electroosmotic flow vectors (b) Streamlines (c) Enhanced liquid mixing

3.2 Multi-electrode model

For an optimal electrode configuration, the present numerical analysis can also be applied to any other viable electrode layout, including number, size, gap, modulating voltages, channel width, and electrolyte properties. Figures 7 and 8 show the velocity vectors and streamlines of the modulated electroosmotic flows in the microchannels with multi-electrodes, including (a) symmetric double electrodes on two walls, (b) double electrodes on one wall, (c) asymmetric double electrodes on two walls, and (d) triple electrodes on two walls. Liquid mixing is shown in Fig. 9. It can be seen that micro-vortices are formed in the microchannel, and liquid mixing is greatly enhanced. The electroosmotic flow rate in the microchannel for the five models is shown in Fig. 10, where Q0 = U0h and U0 = −ε2ε0E0ζ0/μ. It can be seen that the flow rate increases with an increase in the modulating electric field Ve. With fixed modulating potential Ve, 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.
https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig7_HTML.gif
Fig. 7

Electroosmotic flow vectors in the microchannel induced by modulating electric fields of multi-electrodes. The electrokinetic diameter κh = 32 and the dielectric layer κδ = 3.2. (a) Symmetric double electrodes on two walls (b) Double electrodes on one walls (c) Asymmetric double electrodes on two walls (d) Triple electrodes on two wall

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig8_HTML.gif
Fig. 8

Flow streamlines in the microchannel induced by modulating electric fields of multi-electrodes. The electrokinetic diameter κh = 32 and the dielectric layer κδ = 3.2. (a) Symmetric double electrodes on two walls (b) Double electrodes on one walls (c) Asymmetric double electrodes on two walls (d) Triple electrodes on two walls

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig9_HTML.gif
Fig. 9

Enhanced liquid mixing in the microchannel by modulating electric fields of multi-electrodes. The electrokinetic diameter κh = 32 and the dielectric layer κδ = 3.2. (a) Symmetric double electrodes on two walls (b) Double electrodes on one walls (c) Asymmetric double electrodes on two walls (d) Triple electrodes on two walls

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig10_HTML.gif
Fig. 10

Flow rate versus field-modulating potential for five different models. The electrokinetic diameter κh = 32 and the dielectric layer κδ = 3.2

https://static-content.springer.com/image/art%3A10.1007%2Fs10544-010-9450-1/MediaObjects/10544_2010_9450_Fig11_HTML.gif
Fig. 11

Liquid mixing coefficient versus field-modulating potential for five models. The electrokinetic diameter κh = 32 and the dielectric layer κδ = 3.2

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.

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