Analysis of electroosmotic flow with periodic electric and pressure fields via the lattice Poisson–Boltzmann method

Applied Mathematical Modelling 37 (2013) 2816–2829

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Analysis of electroosmotic flow with periodic electric and pressure fields via the lattice Poisson–Boltzmann method Tung-Yi Lin, Chieh-Li Chen ⇑ Department of Aeronautics and Astronautics, National Cheng Kung University, Tainan, Taiwan

a r t i c l e

i n f o

Article history: Received 17 February 2012 Received in revised form 28 May 2012 Accepted 13 June 2012 Available online 21 June 2012 Keywords: Electroosmotic flow Microchannel Lattice Poisson–Boltzmann method Phase angles Heterogeneous surface

a b s t r a c t This paper analyzes the electroosmotic flow fields in heterogeneous microchannels by applying the lattice Poisson–Boltzmann equation. The influences of surface potential, ionic molar concentration, channel height, and driving force fields on fluid velocity are discussed in detail. A scheme for producing vortexes in a straight channel by adjusting the heterogeneous surface potentials and phase angles of the periodic driving force fields is introduced. By distributing the heterogeneous surface potentials at particular positions, we can create vortexes near walls or in the center of the channel. The size, strength, and rotational direction of vortexes are further variable by introducing appropriate phase angles for a single driving force field or for the phase differences between combined driving force fields, such as electric/pressure fields. These obstacle-like vortexes perturb fluids and hinder flow, and thus, may be useful for enhancing micromixer performance. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction In the past decades, micro-fabrication methods that exploit electrokinetics have been developed extensively. These methods have applications in many fields, including microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS). Generally, electrokinetic phenomena are involved in electroosmotic flow (EOF) [1–7], streaming potential [8,9], sedimentation potential [10,11], and electrophoresis [12–15]. Microfluidic applications such as microchannels, micromixers, micropumps, and micronozzles often exploit EOF. Yan et al. [16] used microparticle image velocimetry (l-PIV) to investigate the wall surface zeta potential (1) and fluid behaviors of electroosmotic flow in microchannels. Horiuchi et al. [17] adopted l-PIV to measure the electroosmotic flow-field driven by a combination of electric and pressure fields in trapezoidal microchannels. Additionally, Sze et al. [18] used a method that combined the Smoluchowski equation with the slope of the current-time relationship to determine the 1 potential in microchannels. In an analytical study, Dutta and Beskok [19] solved the problem of a two-dimensional electroosmotic flow driven by both electric and pressure fields, and investigated the associated velocity distribution, mass flow rate, shear stress, and vortices. Sadeghi et al. [20] calculated an approximate solution to electroosmotic flow between parallel plates, and described the effects of the Nusselt number and Debye Hückel parameter on electroosmotic flow. In research concerning numerical simulations, Yau et al. [21] solved macroscopic governing equations by applying a coordinate transformation, and discussed the flow characteristics of electroosmotic flow in microchannels with wavy surfaces. By developing a semi-implicit multi-grid algorithm, Alam and Bowman [22] calculated the divergence-free velocity at grid points inside the electroosmotic flow field. Additionally, Patankar and Hu [23] employed the SIMPLER algorithm to compute precisely the Navier–Stokes equations and the Poisson–Boltzmann equation, and thus, provide a detailed description of flow properties. Hadigol et al. [24] used a finite volume method to simulate the fluid-flow power law in microchannels and micropumps subjected to mixed pressure and electric fields. ⇑ Corresponding author. E-mail address: [email protected] (C.-L. Chen). 0307-904X/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.apm.2012.06.032

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Recently, the lattice Boltzmann method (LBM) was proposed as an alternative numerical scheme for simulation of fluid flows. A mesoscopic statistical numerical algorithm was used to simulate fluid behaviors by tracking the evolution of particle distribution. In their investigation into electroosmotic flows, Wang et al. [25,26] developed the lattice Poisson–Boltzmann method (LPBM) to calculate the electric potential field and fluid flow in a straight microchannel, and discussed mixing performance. Their analytical solutions were in good agreement with the observed electroosmotic flow behaviors, and thus, demonstrated the validity of the LPBM approach. Furthermore, Wang and Kang [27] described the distribution of ionic molar concentrations in the microchannel. For homogeneously charged long channels, the Poisson–Boltzmann model is applicable to a wide range of electric double-layer thicknesses, until the layers fully overlap. Wang showed that the Poisson–Boltzmann model fails to predict the electric potential at high zeta potentials, and applied an enhanced velocity field to heterogeneously charged microchannels. Thus, we provide a validation of the Poisson–Boltzmann model application to our own work; the details are described in Section 2.3. Tang et al. [28] studied EOF mixing enhancement in microchannels using a numerical model similar to that of the LPBM; however, in Tang’s model, both driving force fields were periodically oscillated. Tang’s findings indicated that an advection chaotic flows, and that different mixing efficiencies are obtainable between heterogeneously charged parallel plates. The validity of the Poisson–Boltzmann model’s application to electroosmotic pulsating flow is discussed in Section 2.3. The validity of the LPBM has led to the development of many EOF applications. Wang et al. analyzed EOF pump performance [29]; they found that finite particle size, bulk ionic concentration, external electric field strength, and surface potentials influence EOF pump performance. Moreover, Wang also simulated roughness and cavitation effects on EOF in rough microchannels [30,31]. The height of roughness and depth of cavitation affect EOF behavior in homogeneously and heterogeneously charged microchannels. Additionally, Wang et al. [32,33] investigated electroosmotically driven flow in porous media. The simulated results in [32] show that for a given porosity, flow rates increase with axis length along the direction of the external electric field, and decrease with the angle between the semi-major axis and the bulk flow direction, if the orientation angle is less than p/2. The numerical results of [33] indicate that electro-osmotic permeability through the granular microporous media increases monotonically with porosity, ionic concentration, pH, and temperature. Shi et al. discussed the effect of the Joule heating effect and viscous dissipation on electroosmotic flow [34]; their results showed that temperature variation has a significant effect within an electric double layer (EDL). Nosrati et al. [35] used the LBM to analyze Newtonian fluid flow driven by mixed electroosmotic and pressure force fields in a slit microchannel. Shi et al. [36] proposed a simple model for electroosmotic flow with thermal effects in a microchannel with a thin double layer. Shi’s results were in good agreement with the conventional complete model. The results also showed the behaviors of fluid flow driven by time-dependent pressure and electroosmotic force fields. Furthermore, Tang et al. [37] used the LBM to investigate non-Newtonian fluid flow through microporous media, driven by pressure and electroosmotic force fields. Recently, Masilamani et al. [38] used a hybrid lattice Boltzmann and finite difference model to investigate transport characteristics for combined electroosmotic- and pressure-driven microflows. This review describes advantages of the LBM, as follows: (1) The adopted equation uses simpler mathematics than the conventional Navier–Stokes equations employ, and is more efficient to compute. (2) The LBM is suitable for analyzing flow fields with complex and fractal geometries [39]. (3) The programs are easy to code. (4) The equation can be used with parallel processing in other applications. In this paper, we adopt the LPBM model [25] to study the flow phenomena involved in electroosmotic flow. The flow properties of a fluid affected by the phase angles of driving fields, electric field strength, ionic molar concentration, and double-layer thickness are discussed. The remainder of this paper is organized as follows: Sections 2.1 and 2.2 present a detailed discussion on the concepts of EOF and the LBM for mass, momentum, and electrical potential distribution; Section 2.3 introduces the proposed simulation model; Section 3 shows the results; and lastly, Section 4 offers a conclusion. 2. Physical and numerical models 2.1. Electroosmotic flow Within a microchannel, a negatively charged wall attracts positively charged ions from a buffer solution, thereby forming an electrical double layer. On the application of a voltage, cations in the diffusion region of the EDL migrate in the direction of the cathode, carrying coordinated water molecules with them. The result is a net flow of solution, the so-called electroosmotic flow, toward the negative electrode. 2.2. The lattice Boltzmann method 2.2.1. The lattice Boltzmann equation for mass and momentum densities The LBM, which originated from lattice gas cellular automata (LGCA) [40,41], is a numerical scheme used in computational fluid dynamics. The method is generally used to simulate fluid flow across a regular grid inside the flow domain.

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The LBM can be viewed as a fluid motion model, from the microscopic viewpoint, includes collision and streaming (or propagation) steps. The Navier–Stokes equation and the lattice Boltzmann equation (LBE) provide an approach to compute macroscopic fluid properties such as density and velocity, as an alternative to conventional methods such as the finite difference method and the finite element method. The LBE is derived from the Boltzmann equation (BE), which is discretized with respect to velocity, space, and time. The BE with external applied body force term can be written as follows:

~ Þf þ ~ ~~f ¼ X; @ t f þ ð~ nr F r n

ð1Þ

where f ¼ f ð~ r; ~ n; tÞ is the single particle distribution function in the phase space ð~ r; ~ nÞ, ~ r is the space vector, ~ n is the microscopic velocity vector, ~ F is the external body force, and X is the collision operator. Because of its complexity, the collision term is difficult to manage, and thus, the Bhatnagar, Gross, and Krook (BGK) approximation for X is often adopted for LBM simulations. Eq. (1) is rewritten as

~ Þf þ ~ ~~f ¼  1 ðf  f eq Þ; @ t f þ ð~ nr F r n

ð2Þ

sm

where sm is the relaxation time, and feq is the Maxwell Boltzmann equilibrium distribution function [28] in the form of

f

eq

# ð~ n ~ uÞ2 ; ¼ exp  2RT ð2pRTÞD=2

q0

"

ð3Þ

where q0, ~ u, and T are the fluid density, velocity, and temperature, respectively, R is the gas constant, and D is the space dimension. The external force term can be addressed by a special treatment as

~ ~ uÞ ~ ~~f ¼ G  ðn  ~ f eq ; Fr n RT

ð4Þ

where ~ G is external body force per unit mass of fluid. In the LBM, the BE is discretized in a specified lattice model as follows:

~ Þfi þ ~ ~~fi ¼  1 ðfi  f eq Þ; @ t fi þ ð~ nr Fr n i

i ¼ 0  8; i ¼ 0  8:

sm

ð5Þ

In this study, we applied the D2Q9 [28] model shown in Fig. 1. The lattice velocity components are

8 9 ð0; 0Þ; i ¼ 0 > > > >   > > < = ði1Þp ði1Þp cos 2 ; sin 2 c; i ¼ 1  4 ~ ei ¼ ; > >   > > > > ði5Þ p ði5Þ p p p : cos þ 4 ; sin 2 þ 4 c; i ¼ 5  8 ; 2

ð6Þ

where c is the particle streaming speed. The Maxwell Boltzmann equilibrium distribution function is given by

" fieq ¼ qwi 1 þ 3

2

~2

#

~ u ð~ ei  ~ uÞ 3u ei  ~ ; þ9  c2 2 c2 c4

8 9 > > < 4=9; i ¼ 0 = wi ¼ 1=9; i ¼ 1  4 : > > : ; 1=36; i ¼ 5  8

ð7Þ

For a two-dimensional flow problem, the lattice Boltzmann BGK (LBGK) mass density equation, employing an external force, can be described as

Fig. 1. Schematic diagram of the D2Q9 lattice model.

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fi ð~ r þ~ ei dt; t þ dtÞ ¼ fi ð~ r; tÞ 

1

sm 

~  G  ð~ ei  ~ uÞ eq fi ð~ fi ð~ r; tÞ  fieq ð~ r; tÞ þ dt r; tÞ; RT

ð8Þ

where dt ¼ dxc is the lattice step, with lattice spacing dx. The non-dimensional relaxation time, sm , is defined as sm ¼ sdtm ¼ 3m dxdt2 þ 0:5, with kinetic viscosity m. The collision and streaming (propagation) steps during the evolution of fluid particle motion are described using Eq. (8), whereas the macroscopic fluid density and velocity are given by

8 9 X > q ¼ fi > > > < = i X : > ei fi > u¼ > > : q~ ;

ð9Þ

i

By applying the Chapman–Enskog expansion (or multiscale analysis) [40], the LBGK equation can be recovered to the Navier–Stokes equation as

q

@~ u ~~ ~ p þ lr2~ ur þ qð~ uÞ ¼ r u þ~ FE; @t

ð10Þ

where p is the pressure, and l is the fluid dynamic viscosity. ~ F E represents the electrical body force vector of a charged fluid, expressed as

  ~ FE ¼ ~ Eint þ ~ F ext þ qe ~ n ~ Bint þ ~ FV ;

ð11Þ

where ~ F ext is the external force acting on the fluid by an external force field, which includes the Lorentz force induced by both the external electric and magnetic fields, respectively. ~ F V is the single equivalent force density caused by the intermolecular attraction. If only the electric field is applied, ~ F ext in Eq. (11) decreases to ~ F ext ¼ qe~ Eext is the external electric field Eext , where ~ strength. Therefore, the external electrical force for the electrokinetic flow in a dilute electrolyte solution is given by

~ /; ~ F E ¼ qe~ Eext  qe r

ð12Þ

where / is the streaming electrical potential induced by the movement of ions in the electrolyte solution, as described by Ernst–Planck Theory. Thus, for the external force in Eq. (8), pressure and electrical terms can be rewritten as

~ ~ ~p þ q ~ ~ ~ G  ð~ n ~ uÞ eq ðr e Eext  qe r/Þ  ðei  uÞ eq fi ¼ fi : RT RT

ð13Þ

Generally, streaming electrical potential dominates electro-viscous effects in a purely pressure driven flow. However, the streaming potential is considerably smaller than the external potential, and may be omitted for cases involving purely electrically driven flows. 2.2.2. The lattice Boltzmann equation for electrical potential distribution Fluid macroscopic density and velocity can be computed by coupling Eqs. (6)–(9), and (13); however, the net charge density of the electrolyte solution must be obtained prior to the calculation. EOF theory related to the electrical potential and ions distribution in bulk solution was mentioned earlier, and must obey the Poisson equation

r2 w ¼ 

qe ; ee0

ð14Þ

where e0 is the permittivity of free space, and e is the non-dimensional dielectric constant for the solution. Based on classic EDL theory, ionic number concentration is described by the equilibrium Boltzmann distribution. The relation between net charge density and ionic number concentration is given by

qe ¼

  X zj ew ; zj enj;1 exp kb T j

ð15Þ

where zj is the valence of the jth ion; nj,1 is the ionic concentration of the jth species; e is the charge on an electron, and kb is the Boltzmann constant. Substituting Eq. (15) into Eq. (14) gives

r2 w ¼ 

  1 X zj ew zj enj;1 exp  : kb T ee0 j

ð16Þ

Eq. (16) is a steady state form of the Poisson equation. We define a part of Eq. (16) as g rhs ¼ ee10

P

j zj enj;1

  z ew exp  kj T . b

We can take a framework similar to that of the LBGK equation for the electrical potential distribution function as

g i ð~ r þ~ ei dt; g; t þ dt; gÞ ¼ g i ð~ r; tÞ 

1

sg

   1 ~ dt; gwi g rhs ; g i ð~ r; tÞ  g eq ð r; tÞ þ 1  i 2sg

ð17Þ

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where gi is the electrical potential distribution function, and sg is the relaxation time for the electrical potential distribution function. The electrical potential equilibrium distribution function is given by

g eq i

8 9 i¼0 > > < 0; =  i w; w  i ¼ 1=6; ¼w i¼14 ; > > : ; 1=12; i ¼ 5  8

ð18Þ

The term dt, g is expressed as

dt; g ¼

dx ; c0

ð19Þ

and can be tuned by varying the time interval. Nonetheless, the non-dimensional relaxation time in the electrical potential evolution equation is given as

sg ¼

3vdt; g þ 0:5; 2dx2

ð20Þ

where v is the electrical potential diffusivity, which for the purposes of this paper is arbitrarily set to unity. The macroscopic electrical potential can be evaluated by the following expression:

w¼

X ðg i þ 0:5g rhs dt; gwi Þ:

ð21Þ

i

The net charge density is obtained by combining Eq. (21) with (15). The evolution Eqs. (17)–(20) can be shown to satisfy the Poisson equation using the Chapman–Enskog expansion technique [40]. 2.3. Simulation models In this paper, we used a symmetric electrolyte solution. According to the equilibrium Boltzmann distribution equation, the ionic number concentration of cations and anions can be determined as follows:

  zew n ¼ n1 exp  : kb T

ð22Þ

By substituting Eq. (22) into Eq. (15), we obtain an expression for the net charge density.

  zew : kb T

qe ¼ zeðnþ  n Þ ¼ 2zen1 sinh

ð23Þ

Hence, by associating Eq. (16) with (23), the non-linear Poisson–Boltzmann equation is obtained.

r2 w ¼

2zen1 sinhðkzewT Þ b

ee0

:

ð24Þ

If the electrical potential is considerably smaller than the molecular internal energy, then the sinhðkzewT Þ term can be approxb imated to kzewT , which gives the linearized Poisson–Boltzmann equation as b

2

r w¼

2z2 e2 n1 w w ¼ ; kd ee0 kb T

ð25Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kd is the Debye length (the thickness of the EDL) defined as kd ¼ nee10zk2beT2 ¼ x1 with the Debye-Hückel parameter, x. The physical model used in this study is shown in Fig. 2. The size of the channel is 4 lm in height and 24 lm in length. The anode and cathode are placed at the entrance and exit of the channel, respectively. Because both electrical and pressure driving force fields considered in this work vary according to a sine function, Eq. (13) should be modified as

~ psinc þ q ~ ~ ~ ðr e Eext sin dÞ  ðei  uÞ eq fi : RT

ð26Þ

We next discuss the validity of the Poisson–Boltzmann model, used to determine the relation between ion distribution and electrical potential. If restraints associated with the Poisson–Boltzmann model should cause it to fail, then the Poisson–Nernst–Planck model might provide an alternative. First, in our study, both the electrical and pressure driving force fields exhibit periodic oscillation. Moreover, if the AC voltage signal frequency is sufficiently low, such that the flow problem can be viewed as being in a quasi-steady state, then we can adopt the Poisson–Boltzmann model. From this perspective, we see that the fluid velocity profiles are separately driven by the electric and pressure fields, as shown in Figs. 3 and 7, respectively [28]. The velocity profiles tend toward a quasi-steady state with increasing period in both cases. Thus, if the frequencies of the driving fields are low enough, the ionic concentration can be viewed as being in a steady state for individual phase angles. This is a reason we validate the Poisson–Boltzmann model in this work. Second, the zeta potential values for electrical field strength and ionic molar concentration (C1) also affect the accuracy of the Poisson–Boltzmann model.

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Fig. 2. Diagram of the physical model.

The results reported in [27] demonstrate the applicability of this model under conditions such as low electric field intensity, low ionic molar concentration, and low absolute values of zeta potential. Lower zeta potential values must be adopted for heterogeneously charged microchannels. Therefore, to apply the Poisson–Boltzmann model, we must exercise care when choosing these parameters. We applied a periodic boundary condition along the x-axis by using a bounce-back scheme to implement no-slip walls. The boundary methods used for the electric field were the same as those used by Wang et al. [25]. 3. Results and discussion 3.1. Homogeneous surface potential 3.1.1. Flows driven by a constant electric field We simulated the EOF field under various conditions by applying the LPBM with a constant electric driving force, and using water as the working fluid in the microchannel. We added a symmetric KCl electrolyte to increase the water’s mobility. Simulations were performed for 1 atm pressure at 25 °C. The physical properties used for water in these simulations were density, q0 = 997.044 kg/m3, and kinetic viscosity, m = 8.9755  107 m2/s. Fig. 3 shows a comparison between the non-dimensional numerical solution to the LPBM with periodic driving forces and the linearized analytical solution. The physical parameters are ionic molar concentration C1 = 104 M, surface potential ws = 1 = 50 mV, and electric field strength Ex = 500 V/m. The calculated non-dimensional numerical results are in good agreement with the linearized analytical solution. Moreover, the flow velocity profile exhibits a plug-like tendency. Accord-

Fig. 3. Comparison of LPBM model with linearized analytical solution.

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Fig. 4a. Velocity profiles of flows for various ionic molar concentrations driven along microchannels by a constant electric field.

Fig. 4b. The mass fluxes for various ionic molar concentrations across a center section of the microchannel.

ing to EDL theory, the fluid inside an EDL attracts a large number of charged ions, the motion of which induces fluid flow when an electric current is applied. Therefore, fluid velocity within the EDL is variable. By contrast, the region outside of the EDL exhibits low variability in the fluid velocity profile because fluid flow outside the EDL is generated by interactions between the fluid particles. The magnitude of the electrically driven force is determined by surface potential, and thus, velocity increases with the absolute value of the surface potential. Increasing electric field strength also raises fluid velocity by producing an electrical force of greater magnitude to accelerate the fluid. For an electroosmotic flow driven by a constant electric field, the ionic

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molar concentration of the solution is another important factor that influences the velocity profile. The Debye length, or thickness of the EDL, decreases with increasing C1. Fig. 4(a) shows velocity and flow profiles for various ionic molar concentrations. For C1 = 107 M, the figure shows a parabola-like profile, rather than the plug-like form seen for other ionic concentrations. In this study, we investigated a range of ionic molar concentrations from 107 to 104 M. Fig. 4(b) shows that

Fig. 5. Velocity profiles of flows driven by a periodic electric field with phase angle d, in microchannels.

Fig. 6. Velocity profiles of flows driven by a periodic pressure field with phase angle c in microchannels.

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the total mass flux initially increases in conjunction with concentration, and then smoothly decreases as concentration continues rising. Previous studies have reported that fluid velocity increases in conjunction with the Debye length and channel height ratio. This implies that fluid flow under an extremely broad channel is close to zero. 3.1.2. Flows driven by periodic electric and pressure fields Fig. 5 shows the velocity profile of fluids inside an EOF field, driven by a periodic electric field with a sinusoidal oscillation. The simulation parameters are C1 = 104 M, ws = 1 = 50 mV, and Ex = 500 V/m. The plot shows that the fluid velocity oscillates, and presents a plug-like profile. The fluid speed reaches a maximum for a phase angle of d = (2n + 1/2)p, n e Z. We observed similar tendencies in the velocity profile over the range of investigated phase angles. The sinusoidally oscillating pressure-driven flow is also presented here for comparison. In this case, the magnitude of the driving force is rP = 1  106 N/m3. Fig. 6 shows that similar results to those of the electrically driven flow were obtained by inputting various pressure phase angles (c), although there is no comparison to the parabola-like velocity profile seen in the electrically driven flow. To control the magnitude and direction of fluid velocities in a microchannel, fluid flow was driven by combining the electrical and pressure force fields. Fig. 7 shows the fluid velocities under periodic force fields of rP = 1  103 N/m3 and Ex = 2000 V/m. Other conditions are the same as those used in Fig. 5. Maximum velocity occurs at d  c = 0 and d  c = p, when both driving forces are parallel. 3.2. Heterogeneous surface potential Simulations of a channel with heterogeneous surface potentials were conducted under conditions of C1 = 104 M, ws1 = 11 = 50 mV, ws2 = 12 = 50 mV, and ws3 ¼ 13 ¼ 122 ¼ 25 mV. The fluid flows were driven only by an electric force field, Ex = 500 V/m. Three cases are described as follows: Case I

y¼H ðws1 ¼ 11 ; x 2 ð0; L=6ÞÞ; ðws2 ¼ 12 ; x 2 ðL=6; L=3ÞÞ ðws1 ¼ 11 ; x 2 ðL=3; L=2ÞÞ; ðws2 ¼ 12 ; x 2 ðL=2; 2L=3ÞÞ ðws1 ¼ 11 ; x 2 ð2L=3; 5L=6ÞÞ; ðws2 ¼ 12 ; x 2 ð5L=6; LÞÞ y¼0 ðws1 ¼ 11 ; x 2 ð0; L=6ÞÞ; ðws2 ¼ 12 ; x 2 ðL=6; L=3ÞÞ ðws1 ¼ 11 ; x 2 ðL=3; L=2ÞÞ; ðws2 ¼ 12 ; x 2 ðL=2; 2L=3ÞÞ ðws1 ¼ 11 ; x 2 ð2L=3; 5L=6ÞÞ; ðws2 ¼ 12 ; x 2 ð5L=6; LÞÞ

Fig. 7. Velocity profiles of flows driven by both periodic electric and pressure fields with phase differences, d  c, in microchannels.

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Case II

y¼H ðws1 ¼ 11 ; x 2 ð0; L=5ÞÞ; ðws2 ¼ 12 ; x 2 ðL=5; 2L=5ÞÞ ðws1 ¼ 11 ; x 2 ð2L=5; 3L=5ÞÞ; ðws2 ¼ 12 ; x 2 ð3L=5; 4L=5ÞÞ ðws1 ¼ 11 ; x 2 ð4L=5; LÞÞ y¼0 ðws1 ¼ 11 ; x 2 ð0; L=5ÞÞ; ðws2 ¼ 12 ; x 2 ðL=5; 2L=5ÞÞ ðws1 ¼ 11 ; x 2 ð2L=5; 3L=5ÞÞ; ðws2 ¼ 12 ; x 2 ð3L=5; 4L=5ÞÞ ðws1 ¼ 11 ; x 2 ð4L=5; LÞÞ

Fig. 8. Velocity contours (top) and streamlines (down) of flows driven by a constant electric field in microchannels with heterogeneous surface potentials of (a) case I, (b) case II, and (c) case III.

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Case III

y¼H ðws1 ¼ 11 ; x 2 ð0; L=6ÞÞ; ðws3 ¼ 13 ; x 2 ðL=6; L=3ÞÞ ðws1 ¼ 11 ; x 2 ðL=3; L=2ÞÞ; ðws3 ¼ 13 ; x 2 ðL=2; 2L=3ÞÞ ðws1 ¼ 11 ; x 2 ð2L=3; 5L=6ÞÞ; ðws3 ¼ 13 ; x 2 ð5L=6; LÞÞ y¼0 ðws1 ¼ 11 ; x 2 ð0; L=6ÞÞ; ðws3 ¼ 13 ; x 2 ðL=6; L=3ÞÞ ðws1 ¼ 11 ; x 2 ðL=3; L=2ÞÞ; ðws3 ¼ 13 ; x 2 ðL=2; 2L=3ÞÞ ðws1 ¼ 11 ; x 2 ð2L=3; 5L=6ÞÞ; ðws3 ¼ 13 ; x 2 ð5L=6; LÞÞ 3.2.1. Flows driven by a constant electric field In this section, we discuss electroosmosis in a heterogeneously charged microchannel, driven by a constant electric field. The calculated results of the experimental cases are shown in Fig. 8. In case I (Fig. 8a), 12 uniformly distributed vortexes are inside the channel. The size and strength of the vortexes are identical, while they have distinct rotational directions. The vortexes are produced by forces induced by opposite, periodically charged surface potentials. By applying the periodic surface potentials in various sequences, as in Case II, the flow field in the channel can be changed significantly. Fig. 8(b) shows that as pairs of symmetric vortexes are produced adjacent to the walls, they are accompanied by accelerated flows that pass through the gap between the vortexes. Another type of vortex pair is found in the center of the channel. Fluid passing these center vortexes is also accelerated. The vortexes produced by the heterogeneous surface potentials provide obstacles to the fluid, restraining the fluid flow.

Fig. 9. Velocity contours of flows driven by a periodic electric field with phase angles of (a) d ¼ p8 , (b) d ¼ 38p, (c) d ¼ 138p, and (d) d ¼ 158p in microchannels of case I.

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Fig. 10. Velocity contours of flows driven by both periodic electric and pressure fields with phase differences, (d  c), of (a) p8 , (b) 38p, (c) 58p, and (d) 78p in microchannels of case I.

Similar flow fields are obtainable by changing the magnitude of the applied surface potentials, rather than by changing the patterns. Fig. 8(c) shows the results of Case III, in which we replace ws2 (case I) with ws3. The flow field is changed, and forms vortex obstacles similar to those of Case II, because of differences between |ws1| and |ws3|. Although distinct outcomes are obtained for Cases I, II, and III, the same vortexes can be produced by locating oppositely charged surface potentials at the channel surfaces. Desired vortex structures can then be produced by careful selection of surface potential. 3.2.2. Flows driven by periodic electric and pressure fields In this section, we consider the fluid flows driven by multi-value phase angles of electric and combined electric-pressure fields. We use Case I as an example. Flows driven by a periodic electric field, with phase angles of p8 , 38p, 138p, and 158p ; as shown in Fig. 9, show a similar tendency to that of the p2 case. The differences are only in the strength and rotational direction of the vortexes. Fig. 9 shows that the size and strength of the d ¼ p8 and 158p vortexes are identical, although they have opposite rotational directions. Similar phenomena are also shown by comparing d ¼ 38p and 138p. The perturbed flow field and the resulting vortexes are the same for inputs of d = h and (2p  h), except for their opposite rotational directions. This phenomenon reoccurs when we interchange ws1 and ws2. The combined electric and pressure driven flows are shown in Fig. 10, where we have included a pressure gradient into the force term, to perturb the fluid more strongly. Thus, the desired mixing performance is adjustable by inputting an appropriate phase difference. This method also provides an approach to mixing problems. 4. Conclusions In this paper, we discuss electroosmotic flow fields, where an external force field drives a fluid contained within an EDL, and we describe simulations conducted by applying the LPBM. From a microscopic view, an LPBM-based approach is suitable

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for calculating microflows. Simulations of flows driven by a constant electric field show that increasing the absolute values of surface potentials, increasing the magnitude of electric fields, and reducing the channel height can enhance fluid velocities. Fluid velocity is also affected by the ionic molar concentration. Lower concentrations produce a thicker EDL, across which fluid velocity displays obvious variation. Fluid velocities at a concentration of C1 = 107 M show the parabola-like profile of a pressure driven flow, rather than the plug-like profile of a general electroosmotic flow. In this study, we also discuss velocity profiles for fluid flows driven by a periodic electric field or a periodic pressure field, or both. The velocity profiles oscillate periodically during each cycle. By choosing appropriate phase differences for the combined electric and pressure driving fields, we can determine whether the fluid at the channel center is accelerated or decelerated. For a micromixer, an important issue is to enhance mixing efficiency by perturbing fluids through active or passive schemes, or both. Because electroosmotic flow fields are closely associated with surface potentials and external force fields, we can alter the properties of the flows simply by introducing heterogeneous surface potentials, rather than by altering channel configurations. The results indicate that vortexes are producible in a straight channel by heterogeneous surface potentials distributed at various positions. The rotational direction of vortexes can be defined by adjusting the magnitude of the electrical potentials. Vortexes are produced near walls, or in the middle of the channel where they provide obstacles to fluid flow. Vortexes can be further controlled by applying a combination of periodic driving forces, that is, the size, strength, and rotational direction of vortexes can also be adjusted by inputting appropriate phase angles for a single driving force field, or the phase differences of combined driving force fields. The simulation results presented in this report shows not only the importance to the performance of micromixers of adjusting surface potentials and driving force fields, but also the potential of the LPBM for application in further studies in the EOF field. Acknowledgments The author would like to thank Prof. C. K. Chen and S.C. Chang for their encouragement and comments on the present work. The work is also partly supported by the National Science Council, Taiwan under the Grant No. NSC-100-2221-E006-104-M Y2. References [1] P. Dutta, A. Beskok, T.C. Warburton, Electroosmotic flow control in complex microgeometries, J. MEMS 11 (2002) 36–44. [2] V. Gnanaraj, V. Mohan, B. 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